Question

Sir i need help in two questions, i am stuck !
1) Calculate the population within a 7mile radius if the radialpopulation density is 6(3+r^2)^1/3 ? Use decimal notation and aswer should be in million.
2) Find the volume of solid obtained by rotating the region enclosed by y=x^2 and y=5x about the line x=0. kindly tell me the upper limit and integrand of it, i will solve then by myself.

Answer 1

The population in a ring of radius r is density * area, so adding up all the rings you get

∫[0,7] 6∛(3+r^2) * 2πr dr
hint: let u = 3+r^2

for the volume, you can use
shells:
∫[0,5] 2πrh dx
= ∫[0,5] 2πx(5x-x^2) dx
discs:
∫[0,25] π(R^2-r^2) dy
= ∫[0,25] π(y - (y/5)^2) dy

Answer 2

done thank you steve