QuestionCalculate the area under the curve $y=\frac{4}{1+x^{2}}$ from $x=-1$ to $x=1$ .

Studdy Solution

STEP 1

Assumptions1. The equation of the curve is $y=\frac{4}{1+x^{}}$ . The shaded region under the curve is bounded by $x=-1$ and $x=1$
3. We are asked to find the area of the shaded region under the curve, which is equivalent to calculating the definite integral of the function from $x=-1$ to $x=1$ .

STEP 2

The area under a curve defined by a function $f(x)$ from $x=a$ to $x=b$ is given by the definite integral of $f(x)$ from $a$ to $b$ . So, we need to calculate the definite integral of the given function from $x=-1$ to $x=1$ .
$Area = \int_{-1}^{1} \frac{4}{1+x^{2}} dx$

STEP 3

This integral can be solved using a standard integral formula. The integral of $\frac{1}{1+x^{2}}$ is $\arctan(x)$ . So, we can rewrite the integral as $Area = \int_{-1}^{1} \frac{1}{1+x^{2}} dx = [\arctan(x)]_{-1}^{1}$

STEP 4

Now, we need to evaluate the arctangent function at $x=1$ and $x=-1$ .
$Area =4 [\arctan(1) - \arctan(-1)]$

STEP 5

The arctangent of1 is $\frac{\pi}{4}$ and the arctangent of -1 is $-\frac{\pi}{4}$ . So, we can substitute these values into the equation.
$Area =4 \left[\frac{\pi}{4} - \left(-\frac{\pi}{4}\right)\right]$

STEP 6

implify the expression inside the brackets.
$Area =4 \left[\frac{\pi}{4} + \frac{\pi}{4}\right]$

STEP 7

Combine the terms inside the brackets.
$Area =4 \left[\frac{2\pi}{4}\right]$

STEP 8

implify the expression.
$Area =4 \times \frac{\pi}{2}$

STEP 9

Finally, calculate the area of the shaded region.
$Area =2\pi$ The area of the shaded region under the curve is $2\pi$ square units.

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