basic integration

Calculus Level 2

$\large \int_{-4}^4 |x| \ dx = ?$

The answer is 16.

2 solutions

Raj Magesh
Dec 21, 2014

First of all, your integral should contain a $dx$ . If so,

$\begin{aligned} \int_{-4}^{4} |x| dx &= \int_{-4}^{0} (-x) dx + \int_{0}^{4} x dx \\ &= \left[\dfrac{-x^2}{2}\right]_{-4}^{0} + \left[\dfrac{x^2}{2}\right]_{0}^{4} \\ &= 8 + 8 \\ &= \boxed{16} \end{aligned}$

A quicker approach is to note the parity of the function as follows:

Since $f(x)=|x|$ is an even function, the integral turns to,

$I=2\cdot \int\limits_0^4 |x|\,dx=2\cdot \int\limits_0^4 x\,dx=2\left[\frac{x^2}{2}\right]_0^4=2\times 8 = \boxed{16}$

Prasun Biswas - 6 years, 3 months ago

Chew-Seong Cheong
Apr 2, 2019

$\begin{aligned} I & = \int_{-4}^4 |x| \ dx & \small \color{#3D99F6} \text{Since the integrand is even} \\ & = 2 \int_0^4 |x| \ dx \\ & = 2 \int_0^4 x \ dx \\ & = x^2 \ \bigg|_0^4 \\ & = \boxed{16} \end{aligned}$

basic integration

The answer is 16.

2 solutions

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