What is the integral from 0 to 3 of (xsqrt(14-x^2)) ?

The integral from 0 to 3 of (xsqrt(14-x^2)) is (14sqrt(14)-5sqrt(5))/3

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integral from 0 to 3 of (xsqrt(14-x^2))

\int_{0}^{3} (x 14 - x^{2}) d x

\frac{14 14 - 5 5}{3}

Decimal

13.73428 \dots

Solution steps

\int_{0}^{3} x 14 - x^{2} d x

Apply u-substitution

: \int_{14}^{5} - \frac{u}{2} d u

= \int_{14}^{5} - \frac{u}{2} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{5}^{14} - \frac{u}{2} d u

Take the constant out:

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

= - (- \frac{1}{2} \cdot \int_{5}^{14} u d u)

Apply the Power Rule

: [\frac{2}{3} u^{\frac{3}{2}}]_{5}^{14}

= - (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{5}^{14})

Simplify

= \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{5}^{14}

Compute the boundaries:

\frac{28 14 - 10 5}{3}

= \frac{1}{2} \cdot \frac{28 14 - 10 5}{3}

Simplify

= \frac{14 14 - 5 5}{3}

\frac{\partial}{\partial θ} (ρ sin (φ) sin (θ))

\int \frac{e ^{2 x}}{e ^{x} + 5} d x

\frac{\partial}{\partial x} (y (- 3 - x))

\int cos (x) 7 5 + sin (x) d x

\frac{\partial}{\partial x} (x y - x^{2} y)

What is the integral from 0 to 3 of (xsqrt(14-x^2)) ?
The integral from 0 to 3 of (xsqrt(14-x^2)) is (14sqrt(14)-5sqrt(5))/3