What is the integral from 0 to 7 of 2piy(49-y^2) ?

The integral from 0 to 7 of 2piy(49-y^2) is (2401pi)/2

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integral from 0 to 7 of 2piy(49-y^2)

\int_{0}^{7} 2 π y (49 - y^{2}) d y

\frac{2401 π}{2}

Decimal

3771.48198 \dots

Solution steps

\int_{0}^{7} 2 π y (49 - y^{2}) d y

Take the constant out:

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

= 2 π \cdot \int_{0}^{7} y (49 - y^{2}) d y

Expand

y (49 - y^{2}) : 49 y - y^{3}

= 2 π \cdot \int_{0}^{7} 49 y - y^{3} 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

= 2 π (\int_{0}^{7} 49 y d y - \int_{0}^{7} y^{3} d y)

\int_{0}^{7} 49 y d y = \frac{2401}{2}

\int_{0}^{7} y^{3} d y = \frac{2401}{4}

= 2 π (\frac{2401}{2} - \frac{2401}{4})

Simplify

2 π (\frac{2401}{2} - \frac{2401}{4}) : \frac{2401 π}{2}

= \frac{2401 π}{2}

\frac{d}{d x} (sin (x^{2} y))

\int (1 - sin (x) + tan (x)) d x

\int \frac{12}{1 - x ^{2}} d x

y = 4 e^{5 x}

f (x) = x^{4} + 2 e^{x}

What is the integral from 0 to 7 of 2piy(49-y^2) ?
The integral from 0 to 7 of 2piy(49-y^2) is (2401pi)/2