zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

integral from 0 to 4 of 4xsqrt(16-x^2)

解答

\int_{0}^{4} 4 x 16 - x^{2} d x

解答

\frac{256}{3}

+1

十进制

85.33333 \dots

求解步骤

\int_{0}^{4} 4 x 16 - x^{2} d x

提出常数:

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

= 4 \cdot \int_{0}^{4} x 16 - x^{2} d x

使用换元积分法

= 4 \cdot \int_{16}^{0} - \frac{u}{2} d u

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

= 4 (- \int_{0}^{16} - \frac{u}{2} d u)

提出常数:

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

= 4 (- (- \frac{1}{2} \cdot \int_{0}^{16} u d u))

使用幂法则

= 4 (- (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{16}))

化简

4 (- (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{16})) : 2 [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{16}

= 2 [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{16}

带入上下限计算后得:

\frac{128}{3}

= 2 \cdot \frac{128}{3}

化简

= \frac{256}{3}

作图

流行的例子

integral from 1 to 5 of (-x^2+6x-5)

\int_{1}^{5} (- x^{2} + 6 x - 5) d x

integral from 0 to 2 of 4xe^{x^2}

\int_{0}^{2} 4 x e^{x^{2}} d x

integral from-1 to 0 of 9x^2(2x^3+4)^2

\int_{- 1}^{0} 9 x^{2} (2 x^{3} + 4)^{2} d x

integral from 0 to 2 of xsqrt(3x^2+4)

\int_{0}^{2} x 3 x^{2} + 4 d x

integral from 0 to 15 of x/(sqrt(36+3x))

\int_{0}^{15} \frac{x}{36 + 3 x} d x