Evaluate iterated integral. ∫03∫12(xy3−x)dydx

Question

Evaluate iterated integral.  ∫03∫12(xy3−x)dydx

Lounctirough · Accepted Answer

Step 1
Integration is the process of finding the anti-derivative function of a given function. The integration in defined limits is known as a definite integral.
Double integral is also known as surface integral. The double integral in defined limits gives the area of the region bounded by the given limits.
Step 2
The integral is calculated by using the standard integration

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C

as follows:-

I = \int_{0}^{3} \int_{1}^{2} (x y^{3} - x) d y d x

= \int_{0}^{3} (x {[\frac{y^{4}}{4}]}_{1}^{2} - x {[y]}_{1}^{2}) d x

= \int_{0}^{3} (x (4 - \frac{1}{4}) - x (2 - 1)) d x

= \int_{0}^{3} (\frac{15}{4} x - x) d x

= \int_{0}^{3} \frac{11}{4} x d x

= {[\frac{11 x^{2}}{8}]}_{0}^{3}

= \frac{99}{8}

Thus the integral is evaluated to be

\frac{99}{8}

.

Evaluate iterated integral. \int_{0}^{3}\int_{1}^{2}(xy^{3}-x)d

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Answer & Explanation

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