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Math  /  Calculus

QuestionFind the area of the region bounded by the graph of f(x)=sin⁡xf(x)=\boldsymbol{\operatorname { s i n }} x and the xx-axis on the interval [−π/3,2π/3][-\pi / 3,2 \pi / 3]. he area is □\square Type an exact answer, using radicals as needed.)

Studdy Solution

STEP 1

What is this asking? Find the area between the curve f(x)=sin⁡x f(x) = \sin x and the x x -axis from x=−π3 x = -\frac{\pi}{3} to x=2π3 x = \frac{2\pi}{3} . Watch out! Don't forget to consider the sign of the sine function over the interval!

STEP 2

1. Define the function and interval
2. Identify the integral setup
3. Calculate the integral
4. Evaluate the definite integral

STEP 3

We're dealing with the function f(x)=sin⁡x f(x) = \sin x and we need to find the area between this curve and the x x -axis.
The interval we're interested in is from x=−π3 x = -\frac{\pi}{3} to x=2π3 x = \frac{2\pi}{3} .

STEP 4

To find the area under the curve f(x)=sin⁡x f(x) = \sin x from x=−π3 x = -\frac{\pi}{3} to x=2π3 x = \frac{2\pi}{3} , we need to set up the definite integral:
∫−π32π3sin⁡x dx\int_{-\frac{\pi}{3}}^{\frac{2\pi}{3}} \sin x \, dx

STEP 5

The antiderivative of sin⁡x \sin x is −cos⁡x-\cos x.
So, when we integrate sin⁡x \sin x , we get:
∫sin⁡x dx=−cos⁡x+C\int \sin x \, dx = -\cos x + C

STEP 6

Now, let's evaluate the definite integral from x=−π3 x = -\frac{\pi}{3} to x=2π3 x = \frac{2\pi}{3} :
[−cos⁡x]−π32π3\left[ -\cos x \right]_{-\frac{\pi}{3}}^{\frac{2\pi}{3}}

STEP 7

Substitute the upper limit x=2π3 x = \frac{2\pi}{3} :
−cos⁡(2π3)=−(−12)=12-\cos\left(\frac{2\pi}{3}\right) = -\left(-\frac{1}{2}\right) = \frac{1}{2}

STEP 8

Substitute the lower limit x=−π3 x = -\frac{\pi}{3} :
−cos⁡(−π3)=−(12)=−12-\cos\left(-\frac{\pi}{3}\right) = -\left(\frac{1}{2}\right) = -\frac{1}{2}

STEP 9

Now, calculate the difference:
12−(−12)=12+12=1\frac{1}{2} - \left(-\frac{1}{2}\right) = \frac{1}{2} + \frac{1}{2} = 1

STEP 10

The area of the region bounded by the graph of f(x)=sin⁡x f(x) = \sin x and the x x -axis on the interval [−π3,2π3][- \frac{\pi}{3}, \frac{2\pi}{3}] is **1**.

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