What is the integral from 0 to a of 4xsqrt(a^2-x^2) ?

The integral from 0 to a of 4xsqrt(a^2-x^2) is (4(a^2)^{3/2})/3

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to a of 4xsqrt(a^2-x^2)

\int_{0}^{a} 4 x a^{2} - x^{2} d x

\frac{4 ( a ^{2} ) ^{\frac{3}{2}}}{3}

Solution steps

\int_{0}^{a} 4 x a^{2} - x^{2} d x

Take the constant out:

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

= 4 \cdot \int_{0}^{a} x a^{2} - x^{2} d x

Apply u-substitution

: \int_{a^{2}}^{0} - \frac{u}{2} d u

= 4 \cdot \int_{a^{2}}^{0} - \frac{u}{2} d u

Take the constant out:

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

= 4 (- \frac{1}{2} \cdot \int_{a^{2}}^{0} u d u)

Apply the Power Rule

: [\frac{2}{3} u^{\frac{3}{2}}]_{a^{2}}^{0}

= 4 (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{a^{2}}^{0})

Simplify

4 (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{a^{2}}^{0}) : - 2 [\frac{2}{3} u^{\frac{3}{2}}]_{a^{2}}^{0}

= - 2 [\frac{2}{3} u^{\frac{3}{2}}]_{a^{2}}^{0}

Compute the boundaries:

- \frac{2}{3} (a^{2})^{\frac{3}{2}}

= - 2 (- \frac{2}{3} (a^{2})^{\frac{3}{2}})

Simplify

= \frac{4 ( a ^{2} ) ^{\frac{3}{2}}}{3}

\int_{- 6}^{1} ∣ x + 4 ∣ d x

\int_{- \frac{1}{3}}^{\frac{1}{3}} (\frac{1}{1 + 9 x ^{2}}) d x

\int_{0}^{π} cot (x) tan (x) d x

\int_{- 3}^{0} x (x + 1) (x + 3) d x

\int_{0}^{\frac{1}{2}} 60 x^{2} (1 - x)^{3} d x

What is the integral from 0 to a of 4xsqrt(a^2-x^2) ?
The integral from 0 to a of 4xsqrt(a^2-x^2) is (4(a^2)^{3/2})/3