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 of x^2sqrt(9-x^2)

解答

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

解答

\frac{81 π}{16}

+1

十进制

15.90431 \dots

求解步骤

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

使用三角换元法

= \int_{0}^{\frac{π}{2}} 81 sin^{2} (u) cos^{2} (u) d u

提出常数:

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

= 81 \cdot \int_{0}^{\frac{π}{2}} sin^{2} (u) cos^{2} (u) d u

使用三角恒等式改写

= 81 \cdot \int_{0}^{\frac{π}{2}} \frac{1 - cos ( 4 u )}{8} d u

提出常数:

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

= 81 \cdot \frac{1}{8} \cdot \int_{0}^{\frac{π}{2}} 1 - cos (4 u) d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 81 \cdot \frac{1}{8} (\int_{0}^{\frac{π}{2}} 1 d u - \int_{0}^{\frac{π}{2}} cos (4 u) d u)

\int_{0}^{\frac{π}{2}} 1 d u = \frac{π}{2}

\int_{0}^{\frac{π}{2}} cos (4 u) d u = 0

= 81 \cdot \frac{1}{8} (\frac{π}{2} - 0)

化简

81 \cdot \frac{1}{8} (\frac{π}{2} - 0) : \frac{81 π}{16}

= \frac{81 π}{16}

作图

流行的例子

sum from n=0 to infinity of ((nx)/6)^n

n = 0 \sum \infty (\frac{n x}{6})^{n}

limit as x approaches+2 of sqrt(2-x)

x \to + 2 lim (2 - x)

derivative f(x)=(2t-1)(3t-4)^{-1}

d er i v a t i v e f (x) = (2 t - 1) (3 t - 4)^{- 1}

(\partial)/(\partial x)(x^3+2xy+y^2)

\frac{\partial}{\partial x} (x^{3} + 2 x y + y^{2})

derivative of 0.015+x

\frac{d}{d x} (0.015 + x)