R is the region bounded by the functions f(x) = √7x + 9 and g(x) = 7x/5 + 13/5. Find the area A of R. Enter an exact answer.
R is the region bounded above by the function f(x) = x + 4 and below by the function g(x) = - 3x/4 - 1 over the interval [a, b] where a = -1 and b = 5. Represent R using the Desmos graph below. Submit your answer to this question by dragging the movable points so that the shaded region represents R.
R is the region bounded by the functions f(x) = 2e^x - 2 and g(x) = x^2 - 6. Find the area of the region bounded by the functions on the interval [-1, 1]. Enter an exact answer.
R is the region bounded by the functions f(x) = 2e^x - 2 and g(x) = x^2 - 5. Find the area of the region bounded by the functions on the interval [-1, 1]. Enter an exact answer.
R is the region bounded by the functions f(x) = 3 - 2cos(x) and g(x) = sin(x). Find the area of the region bounded by the functions on the interval [0, π/2]. Enter an exact answer.
R is the region bounded by the functions f(x) = 1 - 3cos(x) and g(x) = 4sin(x). Find the area of the region bounded by the functions on the interval [0, π/2]. Enter an exact answer.
Write the integral required to find the area of the region pictured below that is bounded by the curves u(y) = √y + 3 and v(y) = y+3/4 when integrating with respect to y. Do not evaluate the integral.
Find the area of the region pictured below that is bounded by the functions u(y) = y + 6 and v(y) = y^2 + 6y + 10. Find the area by integrating along the y-axis. Enter your answer as an exact answer.
A factory selling earphones has a marginal cost function MC(x) = 0.003x^2 - 3.333x + 333 and a marginal revenue function given by MR(x) = 333 - 3x, where x represents the number of earphones. MC(x) and MR(x) are in dollars per unit. Find the total profit, or area between the graphs of these curves, between x = 0 and the first intersection point of these curves with x > 0. Enter your answer in dollars, rounded to two decimal places if needed.
A factory selling iPads has a marginal cost function MC(x) = 0.003x^2 - 3.351x + 351 and a marginal revenue function given by MR(x) = 351 - 3x, where x represents the number of iPads. MC(x) and MR(x) are in dollars per unit. Find the total profit, or area between the graphs of these curves, between x = 0 and the first intersection point of these curves with x > 0. Enter your answer in dollars, rounded to two decimal places if needed.
Find the area of the region pictured below that is bounded by the functions u(y) = y + 7 and v(y) = y^2 + 2y - 13. Find the area by integrating along the y-axis. Enter your answer as an exact answer.
Write the integral required to find the area of the region pictured below that is bounded by the curves u(y) = 3√y + 3 and v(y) = y + 3 when integrating with respect to y. Do not evaluate the integral.
A factory selling earphones has a marginal cost function MC(x) = 0.003x^2 - 3.339x + 339 and a marginal revenue function given by MR(x) = 339 - 3x, where x represents the number of earphones. MC(x) and MR(x) are in dollars per unit. Find the total profit, or area between the graphs of these curves, between x = 0 and the first intersection point of these curves with x > 0. Enter your answer in dollars, rounded to two decimal places if needed.
A factory selling earphones has a marginal cost function MC(x) = 0.003x^2 - 3.387x + 387 and a marginal revenue function given by MR(x) = 387 - 3x, where x represents the number of earphones. MC(x) and MR(x) are in dollars per unit. Find the total profit, or area between the graphs of these curves, between x = 0 and the first intersection point of these curves with x > 0. Enter your answer in dollars, rounded to two decimal places if needed.
R is the region bounded by the functions f(x) = 4x^2/3 + 19x/2 + 16 and g(x) = - x^2/4 - 19x/6 - 3. Find the area A of R. Enter an exact answer.
R is the region bounded by the functions f(x) = 4x^2/3 + 19x/2 + 16 and g(x) = - x^2/4 - 19x/6 - 3. Find the area A of R. Enter an exact answer.
Find the area of the region pictured below that is bounded by the functions u(y) = y + 1 and v(y) = y^2 + 2y - 1. Find the area by integrating along the y-axis. Enter your answer as an exact answer.
Define R as the region that is bounded by the graph of the function f(x) = -2e^x/2, the x- axis, x = -1, and x = 0. Use the disk method to find the volume of the solid of revolution when R is rotated around the x axis. Submit an exact answer in terms of π.
Define Q as the region that is bounded by the graph of the function g(y) = -2√y - 1, the y-axis, y = 3, and y = 5. Use the disk method to find the volume of the solid of revolution when Q is rotated around the y axis. Submit an exact answer in terms of π.
Define Q as the region that is bounded by the graph of the function g(y) = 2√y + 1, the y-axis, y = 1, and y = 3. Use the disk method to find the volume of the solid of revolution when Q is rotated around the y axis. Submit an exact answer in terms of π.
Define Q as the region that is bounded by the graph of the function g(y) = 2√y + 2, the y-axis, y = 1, and y = 2. Use the disk method to find the volume of the solid of revolution when Q is rotated around the y axis. Submit an exact answer in terms of π.
Define R as the region that is bounded by the graph of the function f(x) = e^x/2, the x-axis, x = -3, and x = -1. Use the disk method to find the volume of the solid of revolution when R is rotated around the x-axis. Submit an exact answer in terms of π.
Define R as the region that is bounded by the graph of the function f(x) = -2e^-x, the x- axis, x = 0, and x = 1. Use the disk method to find the volume of the solid of revolution when R is rotated around the x-axis. Submit an exact answer in terms of π.
Define R as the region that is bounded by the graph of the function f(x) = -3√3sin(x), the x-axis, x = π/6 , and x = π. Use the disk method to find the volume of the solid of revolution when R is rotated around the x-axis. Submit an exact answer in terms of π.
Define R as the region that is bounded by the graph of the function f(x) = x^3/4 + 1, the x- axis, x = -1, and x = 2. Use the disk method to find the volume of the solid of revolution when R is rotated around the x axis. Submit an exact answer in terms of π.
Define R as the region bounded by the graphs of the function f(x) = x^3/6 , the x-axis, x = 0, and x = 2. Write the definite integral that describes the volume of the solid created by rotating R around the x-axis. You may submit your answer in unsimplified form.
Define R as the region bounded by the functions f(x) = √x + 1 and g(x) = 1 between x = 1 and x = 5. If R is rotated around the x-axis, what is the volume of the resulting solid? Submit an exact answer in terms of π.
Find the volume of the region bounded above by the function f(x) = - x/2 + 3 and below by g(x) = e^x / 4 between x = 0 and x = 2 if the region is rotated around the x-axis. Submit an exact answer in terms of e and π.
Define Q as the region bounded by the functions u(y) = y^2 / 3 + 1 and v(y) = 1 between y = 3 and y = 4. Choose the integral below that describes the volume of the solid created by rotating Q around the line x = -1. ∫3 4 π [(2)^2 - ( y^2 / 3 + 2)^2 ] dy ∫3 4 π [( y^2 / 3 ) 2 - (0)^2 ] dy ∫3 4 π [ y^2 / 3 + 1]^2 dy ∫3 4 π [( y^2 / 3 + 2)^2 - (2)^2 ] dy ∫4 3 π [( y^2 / 3 + 2)^2 - (2)^2 ] dy
Define Q as the region bounded by the functions u(y) = y + 1 and v(y) = y^2 / 4 + 1 between y = 2 and y = 3. If Q is rotated around the y-axis, what is the volume of the resulting solid? Submit an exact answer in terms of π.
Define R as the region bounded by the functions f(x) = x^2 / 3 + 1 and g(x) = x between x = 1 and x = 2. If R is rotated around the x-axis, what is the volume of the resulting solid? Submit an exact answer in terms of π.
Define R as the region bounded by the functions f(x) = √x and g(x) = 1 x between x = 1 and x = 3. Choose the integral below that describes the volume of the solid created by rotating R around the x-axis. ∫1 3 [√x – 1/x ] 2 dx ∫1 3 π [(√x)^2 - ( 1/x )^2 ] dx ∫3 1 π [(√x)^2 - ( 1/x )^2 ] dx ∫1 3 π [( 1/x )^2 - (√x)^2 ] dx ∫1 3 [ 1/x - √x]^2 dx
Define R as the region bounded above by the function f(x) = 2x + 1 and below by g(x) = x + 1 between x = 0 and x = 2. If R is rotated around the line y = -1, what is the volume of the resulting solid? Submit an exact answer in terms of π.
Define Q as the region bounded by the functions u(y) = 1 - 2y and v(y) = y^2 + 1 between y = -2 and y = 0. If Q is rotated around the line x = 1, what is the volume of the resulting solid? Submit an exact answer in terms of π.
Define Q as the region bounded by the functions u(y) = 2y + 1 and v(y) = y^2 + 1 between y = 0 and y = 2. If Q is rotated around the line x = -1, what is the volume of the resulting solid? Submit an exact answer in terms of π.
Q is the region bounded by the graph of v(y) = 5y, x = 4, and y = 0. Find the volume of the solid of revolution formed by revolving Q around the x-axis. Submit an exact answer in terms of π.
Define R as the region bounded by the graphs of f(x) = 3√x and g(x) = x^2 / 4 over the interval [1, 4]. Which of the following represents the volume of the solid of revolution formed by rotating R about the line x = -1 ∫1 4 2π(x + 1) (3√x – x^2 / 4 ) dx ○ ∫1 4 2π(x - 1) ( x^2 / 4 - 3√x) dx ∫1 4 2π(x - 1) (3√x – x^2 / 4 ) dx ∫1 4 2π(x + 1) ( x^2 / 4 - 3√x) dx ∫4 1 2π(x + 1) (3√x – x^2 / 4 ) dx
R is the region bounded above by the graph of f(x) = 6x - 2x^2 and below by the x-axis over the interval [0, 3]. Find the volume of the solid of revolution formed by revolving R around the y-axis. Submit an exact answer in terms of π.
Q is the region bounded by the graph of v(y) = 4y, x = 5, and y = 0. Find the volume of the solid of revolution formed by revolving Q around the x-axis. Submit an exact answer in terms of π.
R is the region bounded above by the graph of f(x) = 2x^2/3 and below by the x-axis over the interval [0, 3]. Find the volume of the solid of revolution formed by revolving R around the y-axis. Submit an exact answer in terms of π.
Define R as the region bounded by the graphs of f(x) = 3x – x^2 and the x-axis. Which of the following represents the volume of the solid of revolution formed by rotating R about the y-axis? ∫0 3 π(3x^3 – x^4 )dx ∫0 3 2π(3x – x^2 )dx ∫0 3 2πxdx ∫0 3 2π(3x^2 – x^3 )dx ∫3 0 2π(3x^2 – x^3 )dx
Q is the region bounded by the graph of v(y) = 6y, x = 2, and y = 0. Find the volume of the solid of revolution formed by revolving Q around the x-axis. Submit an exact answer in terms of π.
Q is the region bounded by the graph of v(y) = 3y, x = 4, and y = 0. Find the volume of the solid of revolution formed by revolving Q around the x-axis. Submit an exact answer in terms of π.
Select the best method to find the volume of a solid of revolution generated by revolving the region bounded by the graph of y = -19x^2 + 19x and the x-axis around the line y = -15. method of cylindrical shells disk method
Select the best method to find the volume of a solid of revolution generated by revolving the region bounded by the graph of y = -17x^2 + 17x and the x-axis around the line y = -13. method of cylindrical shells disk method
Select the best method to find the volume of a solid of revolution generated by revolving the region bounded by the graphs of y = 12x, y = -x + 2, and the x-axis around the x- axis. method of cylindrical shells disk method
Select the best method to find the volume of a solid of revolution generated by revolving the region bounded by the graph of y = -9x^2 + 9x and the x -axis around the line y = -20. method of cylindrical shells disk method
Set up an integral that represents the length of the curve given below over the interval y = 3 to y = 7. g(y) = 5y^2 + 3y - 4 Do not evaluate the integral.
Let f(x) = x/3 - 3. Calculate the arc length of the graph of x = g(y) over the interval x = 3 to x = 7. Enter an exact answer.
Let f(x) = x^3/6 + 6 + 1/2x. Calculate the arc length of the graph of f(x) over the interval [2, 4]. Enter an exact answer.
Set up an integral that represents the length of the curve given below over the interval [3, 7]. f(x) = x^2 - 2x + 5 Do not evaluate the integral.
Let g(y) = y^4/8 - 2 + 1/4y2. Calculate the arc length of the graph of g(y) over the interval [2, 3]. Enter an exact answer.
Let g(y) = y^3/6 + 1 + 1/2y. Calculate the arc length of the graph of g(y) over the interval [2, 4]. Enter an exact answer.
Let f(x) = 2x^3/2. Calculate the arc length of the graph of f(x) over the interval [1, 2]. Enter an exact answer.
Let f(x) = 10x^3/2. Calculate the arc length of the graph of f(x) over the interval [0, 2]. Enter an exact answer.
Let f(x) = x - 3 over the interval [3, 5]. Find the surface area of the surface generated by revolving the graph of f(x) around the x-axis. Enter an exact answer.
Set up an integral that represents the surface area of the surface generated by revolving the graph of f(x) = √6x - 2 around the x-axis over the interval [1, 3]. Do not evaluate the integral.