Find the volume under the graph of $z = 5 - x ^ { 2 }$ over the triangle $0 \leq x \leq 3$ and $0 \leq y \leq 3 - x$ .
A)15.75
B) 42
C) 22.5
D) 7.5
E) 29.25

Question

Find the volume under the graph of $z = 5 - x ^ { 2 }$ over the triangle $0 \leq x \leq 3$ and $0 \leq y \leq 3 - x$ .  
A)15.75
B) 42
C) 22.5
D) 7.5
E) 29.25

Quizplus · Accepted Answer

The volume is given by the integral$$\int_0^3 \int_0^{3-x} (5-x^2) \ dy \ dx$$$$= \int_0^3 \left[ 5y - xy^2 \right]_0^{3-x} \ dx$$$$= \int_0^3 (5(3-x) - x(3-x)^2) \ dx$$$$= \int_0^3 (15 - 5x - 9x^2 + 3x^3) \ dx$$$$= \left[ 15x - \frac{5x^2}{2} - 3x^3 + \frac{3x^4}{4} \right]_0^3$$$$= 15(3) - \frac{5(3)^2}{2} - 3(3)^3 + \frac{3(3)^4}{4}$$$$= 45 - 22.5 - 81 + 67.5 = 15.75$$

Find the Volume Under the Graph Of z=5−x2z = 5 - x ^ { 2 }z=5−x2

Find the Volume Under the Graph Of $z = 5 - x ^ { 2 }$