Question

Math

Posted 6 months ago

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The shaded region is bounded by the graph of the function
$f(x)=\frac{1}{x^{2}+1} \text {, }$
the line $x=k$, and the two coordinate axes.
If the region has area $\pi / 3$, what is the exact value of $k$ ?
```

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

We need to find the value of $k$ such that the area under the curve $f(x) = \frac{1}{x^2 + 1}$ from $x = 0$ to $x = k$ is $\frac{\pi}{3}$. This can be expressed as the definite integral:
$\int_0^k \frac{1}{x^2 + 1} \, dx = \frac{\pi}{3}$

step 2

The integral of $\frac{1}{x^2 + 1}$ is $\arctan(x)$. Therefore, we can rewrite the integral as:
$\left[ \arctan(x) \right]_0^k = \frac{\pi}{3}$

step 3

Evaluating the integral at the bounds, we get:
$\arctan(k) - \arctan(0) = \frac{\pi}{3}$

step 4

Since $\arctan(0) = 0$, the equation simplifies to:
$\arctan(k) = \frac{\pi}{3}$

step 5

Solving for $k$, we take the tangent of both sides:
$k = \tan\left(\frac{\pi}{3}\right)$

step 6

We know that $\tan\left(\frac{\pi}{3}\right) = \sqrt{3}$, so:
$k = \sqrt{3}$

Answer

$k = \sqrt{3}$

Key Concept

Definite Integral

Explanation

The definite integral of a function over an interval gives the area under the curve of that function over that interval. In this problem, we used the integral of $\frac{1}{x^2 + 1}$ to find the value of $k$ that results in a specific area.

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