The graph of the equation x 2/3 + y 2/3 = 2 2/3 is one of a family of curves called astroids because of their starlike appearance (as shown in the accompanying figure). Find the length of this particular astroid by finding the length of half the first-quadrant portion, y = (2 2/3 - x 2/3) 3/2 , √2/2 ≤ x ≤ 2, and multiplying by 8. The length of the given astroid is (Simplify your answer. Type an exact answer, using radicals as needed.)
Find the lateral (side) surface area of the cone generated by revolving the line segment y = 9/7x, 0 ≤ x ≤ 4, about the x-axis. Check your answer with the following geometry formula. Lateral surface area = 1/2×base circumference × slant height Set up the integral that gives the surface area of the cone. (Type exact answers, using π as needed.) The lateral surface area is (Type an exact answer, using π as needed.)
Find the length of the following curve. If you have a grapher, you may want to graph the curve to see what it looks like. y = 2/3(x^2 + 1)^3/2 from x = 3 to x = 6 The length of the curve is. (Type an exact answer, using radicals as needed.)
Find the area of the surface generated by revolving the curve x = y^3/2, 0 ≤ y ≤ 3, about the y-axis. The area of the surface generated by revolving the curve x = y^3/2, 0 ≤ y ≤ 3, about the y- axis is square units. Type an exact answer in terms of π.)
Approximate the arc length of one-quarter of the unit circle (which is π/2) by computing the length of the polygonal approximation with n = 4 segments (see accompanying figure). The length of the line segment on the interval 0 ≤ x ≤ 0.25 is (Round to four decimal places as needed.) The length of the line segment on the interval 0.25 ≤ x ≤ 0.5 is (Round to four decimal places as needed.) The length of the line segment on the interval 0.5 ≤ x ≤ 0.75 is (Round to four decimal places as needed.) The length of the line segment on the interval 0.75 ≤ x ≤ 1 is (Round to four decimal places as needed.) The length of the polygonal approximation with n = 4 segments is (Round to four decimal places as needed.)
Find the length of the curve x = y^4/4 + 1/8y^2 from y = 1 to y = 4. The length of the curve is (Type an integer or a simplified fraction.)
Do the following for the curve y = 2x^2, -3 ≤ x ≤ 3. a. Set up an integral for the length of the curve. b. Graph the curve to see what it looks like. c. Use your grapher’s or computer’s integral evaluator to find the curve's length numerically.
a. Set up an integral for the area of the surface generated by revolving the curve y = tan(x), π/5 ≤ x ≤ π/4 , about the x-axis. b. Graph the curve. c. Use technology to find the area of the surface numerically. a. Set up an integral for the area of the surface. Select the correct choice below and fill in any answer boxes within your choice. (Type exact answers, using π as needed. Use parentheses to clearly denote the argument of each function.) b. Use technology to graph the curve. Choose the correct graph below. c. The area of the surface is square units. (Do not round until the final answer. Then round to the nearest hundredth as needed.)
Find the area of the surface generated by revolving x = 2√9 - y, 0 ≤ y ≤ 35/4 about the y- axis. Set up the integral that gives the area of the given surface. The area is (Type an exact answer, using π as needed.)
Find the area of the surface generated by revolving the curve y = √2x – x^2, 0.75 ≤ x ≤ 1, about the x-axis. Set up the integral that gives the area of the given surface. The area of the surface is square units. (Type an exact answer, using π as needed.)
Find the area of the surface generated by revolving the curve x = e^y + e^-y / 2 in the interval 0 ≤ y ≤ ln 7 about the y-axis. Set up the integral that gives the area of the given surface. The area is (Type an exact answer, using π as needed.)
a. Find a curve with a positive derivative through the point (1,1) whose length integral is given below. b. How many such curves are there? Give reasons for your answer. L = ∫ 4 1 √1 + 1/4x dx a. Let the curve be y = f(x). Determine (dy/dx)^2. (dy/dx)^2 = Determine dy/dx. Dy/dx = ± Because the curve is to have a positive derivative, an equation of the curve is y = As x increases from the lower limit to the upper limit, the curve runs from to (Type ordered pairs.) b. How many such curves are there? Explain. Choose the correct answer below. A. Only one. The function from the previous step is a constant function. B. None. The function from the previous step is not defined at the given x-value. C. Infinitely many. Any arbitrary constant can be added to the function from the previous step. D. Only one. The derivative of the function is known, as well as the value of the function at one value of x.
Find the area of the surface generated by revolving the curve y = x^3/9, 0 ≤ x ≤ 2, about the x-axis. Set up the integral that gives the area of the given surface. (Type exact answers, using π as needed.) The area of the surface is (Type an exact answer, using π as needed.)
A schematic drawing of a 78-ft dome used by a weather service to house radar is shown to the right. a. How much outside surface is there to paint (not counting the bottom)? b. Express the answer to the nearest square foot. a. Set up the integral that can be used to find the amount of surface to paint. Select the correct choice below and fill in the answer box to complete your choice. The outside surface to paint is square feet. (Type an exact answer, using π as needed.) b. The outside surface to paint is approximately square feet. (Round to the nearest integer as needed.)