Find the volume of the indicated region.
-the solid cut from the first octant by the surface $z = 100 - x ^ { 2 } - y$
A) $\frac { 80000 } { 3 }$
B) $\frac { 200000 } { 9 }$
C) $\frac { 100000 } { 3 }$
D) $\frac { 400000 } { 9 }$

Question

Quizplus · Accepted Answer

The limits of integration are $0 \le x \le 10$ and $0 \le y \le 100 - x^2$. So, the volume is$$\int_0^{10} \int_0^{100-x^2} (100 - x^2 - y) \ dy \ dx = \int_0^{10} \left[ 100y - x^2y - \frac{y^2}{2} \right]_0^{100-x^2} \ dx$$$$= \int_0^{10} \left[ 100(100-x^2) - x^2(100-x^2) - \frac{(100-x^2)^2}{2} \right] \ dx$$$$= \int_0^{10} \left[ 10000 - 200x^2 + x^4 - 10000 + 2000x^2 - 5000x^2 + \frac{x^4}{2} \right] \ dx$$$$= \int_0^{10} \left[ -\frac{3000}{2}x^2 + \frac{3}{2}x^4 \right] \ dx$$$$= \left[ -1000x^3 + \frac{1}{2}x^5 \right]_0^{10} = -1000(10)^3 + \frac{1}{2}(10)^5 = \frac{80000}{3}$$

Find the Volume of the Indicated Region z=100−x2−yz = 100 - x ^ { 2 } - yz=100−x2−y

Find the Volume of the Indicated Region $z = 100 - x ^ { 2 } - y$