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

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

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

\int_{0}^{4} π (y + 3)^{2} d y

76 π

Decimal

238.76104 \dots

Solution steps

\int_{0}^{4} π (y + 3)^{2} d y

Take the constant out:

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

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

Expand

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

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

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} y d y + \int_{0}^{4} 6 y d y + \int_{0}^{4} 9 d y)

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

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

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

= π (8 + 32 + 36)

Simplify

= 76 π

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