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.

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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