Question

double Integration integration 0to 1 integration 0x xydydx=?​

Answer 1

Answer:

A double integral is something of the form. ∫ ∫. R ... 57. 0.4 Example. Evaluate. ∫ π/2. 0. ∫ 1. 0 y sinx dy dx. Solution. integral = ∫ π/2. 0 ... from −1 to y, and then y goes from 0 to 1.

Answer 2

Topic:

Double integration

Formula used:
$\displaystyle\int x^n\ dx = \dfrac{x^{n+1}}{n+1} + C$

Solution:

We need to evaluate the following integral.

$\displaystyle\int_0^1\int_0^x (xy)\ dy \ dx$

This iterated integral needs to be firstly evaluated w.r.t. y keeping x as a constant value.

$\displaystyle\longrightarrow\int_0^1x \int_0^x y\ dy \ dx$

$\displaystyle\longrightarrow\int_0^1x \cdot\bigg[ \dfrac{y^2}{2}\bigg]^x_0 \ dx$

$\displaystyle\longrightarrow\int_0^1x \cdot \bigg[\dfrac{x^2}{2} - \dfrac{0^2}{2}\bigg] \ dx$

$\displaystyle\longrightarrow\int_0^1x \cdot \dfrac{x^2}{2} \ dx$

$\displaystyle\longrightarrow\int_0^1 \dfrac{x^3}{2} \ dx$

$\displaystyle\longrightarrow\bigg[ \dfrac{x^{3+1}}{2\cdot (3+1)} \bigg]^1_0$

$\displaystyle\longrightarrow\bigg[ \dfrac{x^4}{8} \bigg]^1_0$

$\displaystyle\longrightarrow\bigg[ \dfrac{1^4}{8} - \dfrac{0^4}{8}\bigg]$

$\displaystyle\longrightarrow\dfrac18$

This is the required answer.