What is the integral from 6 to 7 of pi(2sqrt(x+2))^2 ?

The integral from 6 to 7 of pi(2sqrt(x+2))^2 is 34pi

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integral from 6 to 7 of pi(2sqrt(x+2))^2

\int_{6}^{7} π (2 x + 2)^{2} d x

34 π

Decimal

106.81415 \dots

Solution steps

\int_{6}^{7} π (2 x + 2)^{2} d x

Take the constant out:

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

= π \cdot \int_{6}^{7} (2 x + 2)^{2} d x

Simplify

(2 x + 2)^{2} : 4 (x + 2)

= π \cdot \int_{6}^{7} 4 (x + 2) d x

Take the constant out:

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

= π 4 \cdot \int_{6}^{7} x + 2 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 4 (\int_{6}^{7} x d x + \int_{6}^{7} 2 d x)

\int_{6}^{7} x d x = \frac{13}{2}

\int_{6}^{7} 2 d x = 2

= π 4 (\frac{13}{2} + 2)

Simplify

π 4 (\frac{13}{2} + 2) : 34 π

= 34 π

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What is the integral from 6 to 7 of pi(2sqrt(x+2))^2 ?
The integral from 6 to 7 of pi(2sqrt(x+2))^2 is 34pi