Evaluate the integral.
-$$\int _ { 0 } ^ { 5 } \int _ { 0 } ^ { \sqrt { 25 - y ^ { 2 } } } \int _ { 0 } ^ { 5 x + 10 y } d z d x d y$$
A) 625
B) 25
C) 125
D) 3125

Question

Quizplus · Accepted Answer

The given integral can be solved by integrating step by step from the innermost integral to the outermost. The innermost integral with respect to $z$ is straightforward and gives $z$ evaluated from $0$ to $5x + 10y$, which simplifies to $5x + 10y$. The next integral with respect to $x$ becomes $\int_0^{\sqrt{25-y^2}} (5x + 10y) dx$, which upon integration gives $\frac{5}{2}x^2 + 10yx$ evaluated from $0$ to $\sqrt{25-y^2}$, simplifying to $\frac{5}{2}(25-y^2) + 10y\sqrt{25-y^2}$. The final integral with respect to $y$ over the interval from $0$ to $5$ involves a bit more work but ultimately yields a result of $625$, after evaluating the polynomial and trigonometric parts correctly. This calculation involves recognizing the geometric interpretation of the bounds and the volume being calculated, which corresponds to a quarter of a cylinder with modifications due to the linear $z$ term.

Evaluate the Integral ∫05∫025−y2∫05x+10ydzdxdy\int _ { 0 } ^ { 5 } \int _ { 0 } ^ { \sqrt { 25 - y ^ { 2 } } } \int _ { 0 } ^ { 5 x + 10 y } d z d x d y∫05​∫025−y2​​∫05x+10y​dzdxdy

Evaluate the Integral $\int _ { 0 } ^ { 5 } \int _ { 0 } ^ { \sqrt { 25 - y ^ { 2 } } } \int _ { 0 } ^ { 5 x + 10 y } d z d x d y$