What is the integral from 1 to 2 of pi(4-1/2 x)^2 ?

The integral from 1 to 2 of pi(4-1/2 x)^2 is pi 127/12

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integral from 1 to 2 of pi(4-1/2 x)^2

\int_{1}^{2} π (4 - \frac{1}{2} x)^{2} d x

π \frac{127}{12}

Decimal Notation

33.24852 \dots

Solution steps

\int_{1}^{2} π (4 - \frac{1}{2} x)^{2} d x

Take the constant out:

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

= π \cdot \int_{1}^{2} (4 - \frac{1}{2} x)^{2} d x

Simplify

(4 - \frac{1}{2} x)^{2} : 16 - 4 x + \frac{1}{4} x^{2}

= π \cdot \int_{1}^{2} 16 - 4 x + \frac{1}{4} 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 (\int_{1}^{2} 16 d x - \int_{1}^{2} 4 x d x + \int_{1}^{2} \frac{1}{4} x^{2} d x)

\int_{1}^{2} 16 d x = 16

\int_{1}^{2} 4 x d x = 6

\int_{1}^{2} \frac{1}{4} x^{2} d x = \frac{7}{12}

= π (16 - 6 + \frac{7}{12})

Simplify

π (16 - 6 + \frac{7}{12}) : π \frac{127}{12}

= π \frac{127}{12}

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What is the integral from 1 to 2 of pi(4-1/2 x)^2 ?
The integral from 1 to 2 of pi(4-1/2 x)^2 is pi 127/12