Find the area under y=4sin(x) and above y=4cos(x) for π2≤x≤3π2.
We can start by graphing the two functions:
The shaded region represents the area we are looking for. To find the bounds of integration, we need to find the intersection points of the two functions:
4sin(x) = 4cos(x)
sin(x) = cos(x)
tan(x) = 1
x = π/4 or 5π/4
Since we are only interested in the area between the two functions for π/2 ≤ x ≤ 3π/2, we need to consider only the intersection point x=5π/4.
So, we can set up the integral as follows:
Area = ∫π/2 5π/4 4sin(x) - 4cos(x) dx
= 4∫π/2 5π/4 (sin(x) - cos(x)) dx
= 4[cos(x) + sin(x)]π/2 5π/4
= 4[(0 + 1) - (1/√2 + 1/√2)]
= 4(1 - √2)/2
= 2(1 - √2)
Therefore, the area under y=4sin(x) and above y=4cos(x) for π/2 ≤ x ≤ 3π/2 is 2(1 - √2).
To find the area between two curves, we need to calculate the difference in the integrals of the two curves.
Given the curves y = 4sin(x) and y = 4cos(x), we need to find the area between them for π/2 ≤ x ≤ 3π/2.
First, let's find the points of intersection between the two curves. To do that, we set the two equations equal to each other:
4sin(x) = 4cos(x)
Divide both sides by 4:
sin(x) = cos(x)
Using the identity sin(x) = cos(π/2 – x), we can write:
sin(x) = sin(π/2 – x)
This means that either x = π/2 – x when both sides are positive, or x = 3π/2 – x when both sides are negative.
For both sides to be positive, we have:
x = π/2 – x
2x = π/2
x = π/4
For both sides to be negative, we have:
x = 3π/2 – x
2x = 3π/2
x = 3π/4
So, the points of intersection are x = π/4 and x = 3π/4.
To find the area between the curves, we need to calculate the integral of the upper curve (4sin(x)) minus the integral of the lower curve (4cos(x)).
The integral of 4sin(x) is -4cos(x), and the integral of 4cos(x) is 4sin(x). We evaluate them between the limits π/2 and 3π/2.
Therefore, the area between the curves is given by:
∫[π/2, 3π/2] (4sin(x) - 4cos(x)) dx
= [ -4cos(x) - 4sin(x) ] evaluated from π/2 to 3π/2
= [ -4cos(3π/2) - 4sin(3π/2) ] - [ -4cos(π/2) - 4sin(π/2) ]
= [ -4(0) - (-4(0)) ] - [ -4(1) - (-4(1)) ]
= 0 - (-8)
= 8
Therefore, the area under y = 4sin(x) and above y = 4cos(x) for π/2 ≤ x ≤ 3π/2 is equal to 8 square units.
To find the area under the curve y = 4sin(x) and above the curve y = 4cos(x) for the given interval π/2 ≤ x ≤ 3π/2, follow these steps:
Step 1: Determine the intersection points
- Set the two functions equal to each other: 4sin(x) = 4cos(x)
- Divide both sides by 4 to simplify: sin(x) = cos(x)
- Simplify further by dividing both sides by cos(x): tan(x) = 1
- Determine the values of x where tan(x) = 1. These values occur at π/4 and 5π/4.
Step 2: Determine the integral limits
- The area under the curve can be found by integrating the difference between the two functions over the interval [π/2, 3π/2]. We need to find the limits of integration based on where the two functions intersect.
- Since tan(x) = 1 at π/4 and 5π/4, we can set these as the limits of integration.
Step 3: Set up the integral
- The area under the curve can be calculated by taking the integral of the difference between the two functions: A = ∫[π/4, 5π/4] (4sin(x) - 4cos(x)) dx
Step 4: Evaluate the integral
- Integrate the function (4sin(x) - 4cos(x)) with respect to x over the interval [π/4, 5π/4].
- Evaluate the integral to find the area under the curve.
By following these steps, you should be able to find the area under the curve y = 4sin(x) and above y = 4cos(x) for the given interval π/2 ≤ x ≤ 3π/2.