Question

Convert the integral below to polar coordinates and evaluate the integral. ∫₀⁴/√² ∫ᵧ√¹⁶⁻ʸ² xydxdy or dθ in each box. Then, enter the limits of integration and evaluate the integral to find the volume. ∫ AB∫ CDA= B= C= D= Volume =

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Answer 1

To convert the integral to polar coordinates, substitute x and y with r*cos(theta) and r*sin(theta) respectively, and determine the limits for r and theta. Then evaluate the integral to find the volume.

Step-by-step explanation:

To convert the given integral to polar coordinates, we need to express the variables x and y in terms of polar coordinates. In polar coordinates, we have x = r*cos(theta) and y = r*sin(theta). The limits of integration for the inner integral will be the lower and upper bounds of y, and for the outer integral, we integrate with respect to r from the lower bound to the upper bound. The integral becomes:

∫04 ∫0sqrt(16-y^2) (x*y) dxdy

Substituting x = r*cos(theta) and y = r*sin(theta), the integral becomes:

∫04 ∫0sqrt(16-y^2) (r*cos(theta)*r*sin(theta)) * (r dr dtheta)

To evaluate the integral, we need to determine the limits for r and theta. The limits for r will be from 0 to 4, and the limits for theta will be from 0 to 2*pi. Evaluating the integral gives the volume:

Volume = ∫02*pi ∫04 ∫0sqrt(16-y^2) (r*cos(theta)*r*sin(theta)) * r dr dy dtheta