Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis. y = 9−x2 y = 0 x = 2 x = 3
A rectangle in the first quadrant has can be made with its lower left vertex at the origin, one side along the x-axis, one side along the y-axis, and upper right vertex on the graph of y = 8−x3. Denote the upper right vertex of the rectangle as (x, y), with x and y measured in meters. a) (3 points) Find a formula for P(x) the perimeter of the rectangle with right side at location x (as a function of x). P(x) = b) (4 points) What is the interval of interest for P(x)? Explain. x in c) (8 points) Find the x-value that gives the maximum perimeter over all such rectangles. (Don't forget to JUSTIFY your answer.)
Consider the following. y = x2 y = 12 − x (i) (a) Find the area of the region by integrating with respect to x. (b) Find the area of the region by integrating with respect to y.
Find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the given lines. y = x y = 0 x = 3 (a) the x-axis (b) the y-axis (c) the line x = 3 (d) the line x = 9
Consider the following function. f(x) = (sin(x))sin(x) (a) Graph the function. (b) Explain the shape of the graph by computing the limit as x→0+. limx→0+ f(x) = (c) Use calculus to find the exact maximum and minimum values of f(x). (If an answer does not exist, enter DNE.) (d) Use a computer algebra system to compute f′′. Then use a graph of f′′ to estimate the x-coordinates of the inflection points. (Round your answer to two decimal places.) smaller value x = larger value x =
Find the volume of the solid obtained by rotating the region bounded by x = 3−y2 and x = −2y about the line x = 4. Round to the nearest thousandth.
A computer algebra system is recommended. Graph the region between the curves. y = cos(x), y = x + 4sin4(x) (i) Compute the area of the region. (Round your answer to five decimal places.)
Use a computer algebra system to find the exact volume of the solid. between the paraboloids z = 2x2 + y2 and z = 7 − x2 − 2y2 and inside the cylinder x2 + y2 = 1
If the curve y = e−x/8 sin(x), x ≥ 0, is rotated about the x-axis, the resulting solid looks like an infinite decreasing string of beads. (a) Find the exact volume of the nth bead. (Use either a table of integrals or a computer algebra system.) (b) Find the total volume of the beads.
Use a computer algebra system to find u×v. u = ⟨5, −9, 8.5⟩ v = ⟨4.5, 5.5, 3.5⟩ Use a computer algebra system to find a unit vector orthogonal to u and v. (Round your answers to four decimal places.)
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 2. y = 1−x x = 0 y = 0
Consider the following. x = 16 − y2 x = y − 4 (a) Find the area of the region by integrating with respect to x. (b) Find the area of the region by integrating with respect to y.
The integrals below give the area of a region in the xy-plane. Sketch the region, label each bounding curve with its equation, and give the coordinates of the points where the curves intersect. Then find the area of the region. ∫0 π4 ∫2sinx 2cosx dy dx Sketch the region, label each bounding curve with its equation, and give the coordinates of the points where the curves intersect. Choose the correct answer below. A. B. C. The area of the region is (Type an exact answer, using radicals as needed.)
Find the area of the region in the first quadrant bounded on the left by the y-axis, below by the curve x = 2 y, above left by the Curve x = (y−1)2, and above right by the line x = 3−y
Find the volume of the solid (by any method) generated by revolving the region bounded by the graphs of the equations about the given axis. y = 1/x, y = 0, x = 3, x = 10 Revolve around the x -axis. a) 4 15π b) 13 30π c) 13 60π d) 730π e) 7 60π
Let V be the region inside the cone x2+y2 = z2, and between the planes z = 1 and z = 2: V = {(x, y, z)∣x2 + y2 ≤ z2, 1 ≤ z ≤ 2} a. Sketch the region of integration R. b. Setup the triple integral in Cartesian coordinates. c. Setup the triple integral in Cylindrical coordinates. d. Setup the triple integral in Spherical coordinates e. Find the volume of V by evaluating an integral in spherical coordinates.
A computer algebra system is recommended. Let R be the region in the first quadrant bounded by the y-axis and the graphs of y = x2 + 3 and y = 8 − ex. (a) Find the area of the region R. (Round your answer to three decimal places.) (b) Find the volume of the solid that results when R is rotated about the x-axis. (Round your answer to three decimal places.) (c) The solid S has base R. Each cross-section perpendicular to the x-axis is a semicircle whose diameter lies in R. Find the volume of the solid S. (Round your answer to three decimal places ∫0 k(8−ex)−(x2+3)dx = ∫ka(8−ex)−(x2+3)dx ∫0 k(8−ex)dx = ∫0 k(x2+3)dx π∫0 k(8−ex)2−(x2+3)2 dx = π∫ka(8−ex)2−(x2+3)2 dx ∫0 a(8−ex)−(x2+3)dx = ∫0 k(8−ex)−(x2+3)dx ∫ka(8−ex)dx = ∫ka(x2+3)dx
Consider the following. y = x2 y = 3 + x (a) Use a graphing utility to graph the region bounded by the graphs of the equations. (b) Use the integration capabilities of the graphing utility to approximate the area to four decimal places.
Sketch the region bounded by the given lines. Then express the region's area as an iterated double integral and evaluate the integral. The coordinate axes and the line 2x + 2y = 4. Sketch the region. Choose the correct graph below. A. B. C. D. Express the shaded area as an iterated integral. Select the correct choice below and fill in the answer boxes to complete your choice. A. A = ∫02∫dydx + ∬dydx B. A = ∬dydx
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = x2, x = y2; about y = 1 V = Sketch the region.
Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y = x, y = xe 1−x/2, about y = 4
The base of a solid is the region bounded by the graphs of y = 4 − x2 and y = 0. The cross sections perpendicular to the y-axis are equilateral triangles. Find the volume of the solid.
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. y = ex/2 (1+ex) 5/2, y = 0, x = 0, x = 3; the x-axis
Find the values of c such that the area of the region bounded by the parabolas y = x2 − c2 and y = c2 − x2 is 72.
Sketch the region bounded by the graphs of the given equations and find the area of that region. y = x2+x−2, y = −ex−2, x = 0, x = 1
Find the volume of the solid obtained by rotating the region bounded by y = 7 x4 and y = 7x about the line y = 7. 1 196 28π 9 π9 196π 15 7 9
Sketch the region bounded by the graphs of the given equations and find the area of their region. x = y2+1, x = y−2, y = −3, y = 3