Problem List Next Problem (1 point) Consider the solid under the graph of 2- e -

Question

Problem List Next Problem (1 point) Consider the solid under the graph of 2- e -, above the disk z2 + y2 0 (a) Set up the integral to find the volume of the solid Instructions: Please enter the integrand in the first answer box, typing theta for D epending on the order of integration you choose, enter dr and d eta in either order into the second and third answer boxes with only one dr or dtheta in each box. Then, enter the limits of integration. B CD A JC (b) Evaluate the integral and find the volume. Your answer will be in terms of a Volume V (c) What does the volume approach as a oo? lim V 400

Explanation / Answer

convert to polar coordinates

x^2 + y^2 < a^2 ==> x ^2 + y^2 = a^2 it is circle of radius a

since we know r^2 = x^2 + y^2 = a^2 , r^2 = a^2 ==> r = + / - a but a > 0

so ==> r = 0 to a

for a circle theta = 0 to 2pi

V(a) = Int{theta = 0 to 2pi} Int{r = 0 to a} e^(-r^2)r drd(theta)

so A = 0 , B = 2 pi ,C = 0 , D = a
= (2pi) (-1/2 e^(-r^2) | 0 to a) ,integrate r and theta independently
= -pi (e^(-a^2) - 1)

V(a) = (1 - e^(-a^2)) pi

Now,
lim a->inf V(a) = lim a->inf (1 - e^(-a^2)) pi = pi

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