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 3/4 of xsqrt(9-4x^2)

解答

\int_{0}^{\frac{3}{4}} x 9 - 4 x^{2} d x

解答

\frac{72 - 27 3}{32}

+1

十进制

0.78858 \dots

求解步骤

\int_{0}^{\frac{3}{4}} x 9 - 4 x^{2} d x

使用换元积分法

= \int_{9}^{\frac{27}{4}} - \frac{u}{8} d u

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

= - \int_{\frac{27}{4}}^{9} - \frac{u}{8} d u

提出常数:

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

= - (- \frac{1}{8} \cdot \int_{\frac{27}{4}}^{9} u d u)

使用幂法则

= - (- \frac{1}{8} [\frac{2}{3} u^{\frac{3}{2}}]_{\frac{27}{4}}^{9})

化简

= \frac{1}{8} [\frac{2}{3} u^{\frac{3}{2}}]_{\frac{27}{4}}^{9}

带入上下限计算后得:

18 - \frac{27 3}{4}

= \frac{1}{8} (18 - \frac{27 3}{4})

化简

= \frac{72 - 27 3}{32}

作图

流行的例子

integral from 0 to 2 of x^2cos(npix)

\int_{0}^{2} x^{2} cos (nπ x) d x

integral from-infinity to 0 of xe^{-6x}

\int_{- \infty}^{0} x e^{- 6 x} d x

integral from 0 to 1-y of y^2

\int_{0}^{1 - y} y^{2} d y

integral from 1 to infinity of 1/(2x^3)

\int_{1}^{\infty} \frac{1}{2 x ^{3}} d x

integral from pi/6 to pi/3 of 3cos(x)

\int_{\frac{π}{6}}^{\frac{π}{3}} 3 cos (x) d x