Question

Integrate f(x,y)=x2 +y over the region bounded by y=2x2 , X-axis , x = 1/2 .

Answer 1

Answer on Question # 41420 – Math – Integral Calculus

Integrate $f(x,y)=x^2 + y$ over the region bounded by $y=2x^2$, $x$-axis, $x = \frac{1}{2}$.

Solution:

$$y = 2x^2, x = \frac{1}{2}, y = 0 - x\text{-axis}.$$

Let sketch the region:

Let Integrate $f(x,y)=x^2 + y$:

$$\begin{array}{l} \int_0^{\frac{1}{2}} dx \int_0^{2x^2} (x^2 + y) dy = \int_0^{\frac{1}{2}} dx \left(x^2 y + \frac{y^2}{2}\right) \int_0^{2x^2} y = \int_0^{\frac{1}{2}} \left(2x^4 + \frac{4x^4}{2}\right) dx = \\ = \int_0^{\frac{1}{2}} 4x^4 dx = \frac{4x^5}{5} \int_0^{\frac{1}{2}} 4x^4 dx = \frac{4x^5}{5} \cdot \frac{1}{32} = \frac{1}{40} \end{array}$$

Answer: $\frac{1}{40}$.

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