Math / Calculus

QuestionFind the shaded region in the graph.
The area of the shaded region is $\square$ . (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type an exact answer in terms of $\pi$ .)

Studdy Solution

STEP 1

1. We are given two functions: $y = 2$ and $y = \cos x + 1$ .
2. The shaded region is the area between these two curves.
3. The intersection point is at $x = \frac{\pi}{4}$ .

STEP 2

1. Determine the points of intersection of the two curves.
2. Set up the integral to find the area of the shaded region.
3. Evaluate the integral to find the exact area.

STEP 3

Determine the points of intersection by setting the equations equal to each other:
$2 = \cos x + 1$

STEP 4

Solve for $x$ :
$2 - 1 = \cos x$ $1 = \cos x$
The cosine function equals 1 at $x = 0$ .

STEP 5

The region of interest is between $x = 0$ and $x = \frac{\pi}{4}$ .
Set up the integral of the difference between the two functions:
$\text{Area} = \int_{0}^{\frac{\pi}{4}} (2 - (\cos x + 1)) \, dx$

STEP 6

Simplify the integrand:
$\text{Area} = \int_{0}^{\frac{\pi}{4}} (2 - \cos x - 1) \, dx$ $\text{Area} = \int_{0}^{\frac{\pi}{4}} (1 - \cos x) \, dx$

STEP 7

Evaluate the integral:
$\text{Area} = \left[ x - \sin x \right]_{0}^{\frac{\pi}{4}}$

STEP 8

Substitute the limits of integration:
$\text{Area} = \left( \frac{\pi}{4} - \sin\left(\frac{\pi}{4}\right) \right) - \left( 0 - \sin(0) \right)$

STEP 9

Simplify the expression:
$\text{Area} = \frac{\pi}{4} - \frac{\sqrt{2}}{2}$
The area of the shaded region is:
$\boxed{\frac{\pi}{4} - \frac{\sqrt{2}}{2}}$

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QuestionFind the shaded region in the graph.
The area of the shaded region is $\square$ . (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type an exact answer in terms of $\pi$ .)

Studdy Solution

STEP 1

1. We are given two functions: $y = 2$ and $y = \cos x + 1$ .
2. The shaded region is the area between these two curves.
3. The intersection point is at $x = \frac{\pi}{4}$ .

STEP 2

1. Determine the points of intersection of the two curves.
2. Set up the integral to find the area of the shaded region.
3. Evaluate the integral to find the exact area.

STEP 3

Determine the points of intersection by setting the equations equal to each other:
$2 = \cos x + 1$

STEP 4

Solve for $x$ :
$2 - 1 = \cos x$ $1 = \cos x$
The cosine function equals 1 at $x = 0$ .

STEP 5

The region of interest is between $x = 0$ and $x = \frac{\pi}{4}$ .
Set up the integral of the difference between the two functions:
$\text{Area} = \int_{0}^{\frac{\pi}{4}} (2 - (\cos x + 1)) \, dx$

STEP 6

Simplify the integrand:
$\text{Area} = \int_{0}^{\frac{\pi}{4}} (2 - \cos x - 1) \, dx$ $\text{Area} = \int_{0}^{\frac{\pi}{4}} (1 - \cos x) \, dx$

STEP 7

Evaluate the integral:
$\text{Area} = \left[ x - \sin x \right]_{0}^{\frac{\pi}{4}}$

STEP 8

Substitute the limits of integration:
$\text{Area} = \left( \frac{\pi}{4} - \sin\left(\frac{\pi}{4}\right) \right) - \left( 0 - \sin(0) \right)$

STEP 9

Simplify the expression:
$\text{Area} = \frac{\pi}{4} - \frac{\sqrt{2}}{2}$
The area of the shaded region is:
$\boxed{\frac{\pi}{4} - \frac{\sqrt{2}}{2}}$

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QuestionFind the shaded region in the graph.The area of the shaded region is □\square□ . (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type an exact answer in terms of π\piπ.)

Studdy Solution

STEP 1

1. We are given two functions: y=2 y = 2 y=2 and y=cos⁡x+1 y = \cos x + 1 y=cosx+1.2. The shaded region is the area between these two curves.3. The intersection point is at x=π4 x = \frac{\pi}{4} x=4π​.

STEP 2

1. Determine the points of intersection of the two curves.2. Set up the integral to find the area of the shaded region.3. Evaluate the integral to find the exact area.

STEP 3

Determine the points of intersection by setting the equations equal to each other:2=cos⁡x+1 2 = \cos x + 1 2=cosx+1

STEP 4

Solve for x x x:2−1=cos⁡x 2 - 1 = \cos x 2−1=cosx 1=cos⁡x 1 = \cos x 1=cosxThe cosine function equals 1 at x=0 x = 0 x=0.

STEP 5

The region of interest is between x=0 x = 0 x=0 and x=π4 x = \frac{\pi}{4} x=4π​.Set up the integral of the difference between the two functions:Area=∫0π4(2−(cos⁡x+1)) dx \text{Area} = \int_{0}^{\frac{\pi}{4}} (2 - (\cos x + 1)) \, dx Area=∫04π​​(2−(cosx+1))dx

STEP 6

Simplify the integrand:Area=∫0π4(2−cos⁡x−1) dx \text{Area} = \int_{0}^{\frac{\pi}{4}} (2 - \cos x - 1) \, dx Area=∫04π​​(2−cosx−1)dx Area=∫0π4(1−cos⁡x) dx \text{Area} = \int_{0}^{\frac{\pi}{4}} (1 - \cos x) \, dx Area=∫04π​​(1−cosx)dx

STEP 7

Evaluate the integral:Area=[x−sin⁡x]0π4 \text{Area} = \left[ x - \sin x \right]_{0}^{\frac{\pi}{4}} Area=[x−sinx]04π​​

STEP 8

Substitute the limits of integration:Area=(π4−sin⁡(π4))−(0−sin⁡(0)) \text{Area} = \left( \frac{\pi}{4} - \sin\left(\frac{\pi}{4}\right) \right) - \left( 0 - \sin(0) \right) Area=(4π​−sin(4π​))−(0−sin(0))

STEP 9

Simplify the expression:Area=π4−22 \text{Area} = \frac{\pi}{4} - \frac{\sqrt{2}}{2} Area=4π​−22​​ The area of the shaded region is:π4−22 \boxed{\frac{\pi}{4} - \frac{\sqrt{2}}{2}} 4π​−22​​​

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QuestionFind the shaded region in the graph.
The area of the shaded region is $\square$ . (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type an exact answer in terms of $\pi$ .)

1. We are given two functions: $y = 2$ and $y = \cos x + 1$ .
2. The shaded region is the area between these two curves.
3. The intersection point is at $x = \frac{\pi}{4}$ .

1. Determine the points of intersection of the two curves.
2. Set up the integral to find the area of the shaded region.
3. Evaluate the integral to find the exact area.

Determine the points of intersection by setting the equations equal to each other:
$2 = \cos x + 1$

Solve for $x$ :
$2 - 1 = \cos x$ $1 = \cos x$
The cosine function equals 1 at $x = 0$ .

The region of interest is between $x = 0$ and $x = \frac{\pi}{4}$ .
Set up the integral of the difference between the two functions:
$\text{Area} = \int_{0}^{\frac{\pi}{4}} (2 - (\cos x + 1)) \, dx$

Simplify the integrand:
$\text{Area} = \int_{0}^{\frac{\pi}{4}} (2 - \cos x - 1) \, dx$ $\text{Area} = \int_{0}^{\frac{\pi}{4}} (1 - \cos x) \, dx$

Evaluate the integral:
$\text{Area} = \left[ x - \sin x \right]_{0}^{\frac{\pi}{4}}$

Substitute the limits of integration:
$\text{Area} = \left( \frac{\pi}{4} - \sin\left(\frac{\pi}{4}\right) \right) - \left( 0 - \sin(0) \right)$

Simplify the expression:
$\text{Area} = \frac{\pi}{4} - \frac{\sqrt{2}}{2}$
The area of the shaded region is:
$\boxed{\frac{\pi}{4} - \frac{\sqrt{2}}{2}}$