Question

Compute the volume by rotating the region in the integrals around the y-axis using a triple integral. Then do compute it over the x-axis.

Answer 1

Evaluate integral_1^2 integral_2x + 2^7 dy dx implies here limits of x are x = 1 to x = 2 that is outside limits & limits of y are y = 2x + 2 to y = 7 since limits of y are variable & limits of x are constants Therefore we evaluate the integral first with respect to y and then respect to x therefore integral_1^2 integral_2x + 2^7 dy dx = integral_1^2 [integral_2x + 2^7 dy] dx = integral_1^2[[y]_2x + 2^7]dx = integral_1^2[7 - (2x + 2)]dx = integral_1^2(7 -2x -2)dx = integral_1^2 (5 -2x)dx = integral_1^2 5 dx + integral_1^2 -2x dx = 5 integral_1^2 dx -2 integral_1^2 x dx = 5[x]_1^2 -2[x^2/2]_1^2 = 5(2 -1) -2/2(2^2 -1^2) = 5 times 1 -1 times (3) = 5 -3 = 2 therefore integral_1^2 integral_2x + 2^7 dy dx = 2