What is the integral from 0 to 4 of pi(sqrt(x)+3)^2 ?

The integral from 0 to 4 of pi(sqrt(x)+3)^2 is 76pi

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integral from 0 to 4 of pi(sqrt(x)+3)^2

\int_{0}^{4} π (x + 3)^{2} d x

76 π

Decimal

238.76104 \dots

Solution steps

\int_{0}^{4} π (x + 3)^{2} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= π \cdot \int_{0}^{4} (x + 3)^{2} d x

Expand

(x + 3)^{2} : x + 6 x + 9

= π \cdot \int_{0}^{4} x + 6 x + 9 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= π (\int_{0}^{4} x d x + \int_{0}^{4} 6 x d x + \int_{0}^{4} 9 d x)

\int_{0}^{4} x d x = 8

\int_{0}^{4} 6 x d x = 32

\int_{0}^{4} 9 d x = 36

= π (8 + 32 + 36)

Simplify

= 76 π

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What is the integral from 0 to 4 of pi(sqrt(x)+3)^2 ?
The integral from 0 to 4 of pi(sqrt(x)+3)^2 is 76pi