shadsiei

2021-01-10

Evaluate the following iterated integrals.

${\int}_{1}^{2}{\int}_{0}^{1}(3{x}^{2}+4{y}^{3})dydx$

irwchh

Skilled2021-01-11Added 102 answers

To evaluate the integral: ${\int}_{1}^{2}{\int}_{0}^{1}(3{x}^{2}+4{y}^{3})dydx$

Solution:

When we integrate with respect to one variable then other will be kept as constant.

Evaluating the integral.

${\int}_{1}^{2}{\int}_{0}^{1}(3{x}^{2}+4{y}^{3})dy={\int}_{1}^{2}[{\int}_{0}^{1}(3{x}^{2}+4{y}^{3})dy]dx$

$={\int}_{1}^{2}[(3{x}^{2}y+4\cdot \frac{{y}^{4}}{4}{)}_{0}^{1}]dx$

$={\int}_{1}^{2}[3{x}^{2}\cdot 1+{1}^{4}-0]dx$

$={\int}_{1}^{2}(3{x}^{2}+1)dx$

$=[3\cdot \frac{{x}^{3}}{3}+x{]}_{1}^{2}$

$=[{x}^{3}+x{]}_{1}^{2}$

$=[({2}^{3}+2)-({1}^{3}+1)]$

$=[12-2]$

$=10$

$\text{Hence, required answer is 10}$

Solution:

When we integrate with respect to one variable then other will be kept as constant.

Evaluating the integral.