Find the area of the region enclosed by the curve y=xsinx and the x-axis for 7π ≤x ≤8π .
what's the trouble? Straightforward integration by parts:
∫[7π,8π] x sinx dx
= sinx - x cosx [7π,8π]
= (0-(8π)(1))-(0-(7π)(-1))
= -15π
What methods have you been taught for this kind of integration ?
Do you know how to do integration by parts ?
To find the area of the region enclosed by the curve y = xsinx and the x-axis for the interval 7π ≤ x ≤ 8π, we need to integrate the absolute value of the function being integrated over this interval.
Step 1: Find the absolute value of the function.
Since the function y = x sin(x) is positive in the given interval, we can work with the function itself.
Step 2: Set up the integral.
The integral of the function y = x sin(x) over the interval 7π ≤ x ≤ 8π is given by:
∫[7π, 8π] x sin(x) dx
Step 3: Evaluate the integral.
To find the integral of x sin(x), we use integration by parts.
Let u = x and dv = sin(x) dx.
Then du = dx and v = -cos(x).
Using the formula for integration by parts: ∫u dv = uv - ∫v du, we have:
∫ x sin(x) dx = -x cos(x) - ∫ (-cos(x)) dx
= -x cos(x) + sin(x) + C
Step 4: Evaluate the definite integral.
Plugging in the limits of integration, we have:
∫[7π, 8π] x sin(x) dx
= [-8π cos(8π) + sin(8π)] - [-7π cos(7π) + sin(7π)]
= [8π + 0] - [-7π + 0]
= 8π + 7π
= 15π
Therefore, the area of the region enclosed by the curve y = x sin(x) and the x-axis for the interval 7π ≤ x ≤ 8π is 15π square units.
To find the area of the region enclosed by the curve y = x*sin(x) and the x-axis for the given interval, we need to integrate the absolute value of the function over that interval.
1. First, we need to find the x-intercepts of the curve by setting y = 0:
0 = x*sin(x)
This gives us two solutions: x = 0 and sin(x) = 0. The latter occurs when x is a multiple of π.
2. Next, we need to determine the limits of integration for the given interval 7π ≤ x ≤ 8π. We know that the function has an x-intercept at x = 0, so we don't need to worry about that.
3. Since we are given the interval in terms of π, we can rewrite it as follows:
7π ≤ x ≤ 8π
This means we need to integrate the function from x = 7π to x = 8π.
4. The absolute value of the function y = x*sin(x) is necessary because the region is enclosed by the curve and the x-axis. Thus, we need to account for the negative values of y.
5. The integral to find the area is:
Area = ∫[7π, 8π] |x*sin(x)| dx
We integrate the absolute value of the function from x = 7π to x = 8π.
6. Finally, to calculate the area, you will need to evaluate the definite integral ∫[7π, 8π] |x*sin(x)| dx using techniques such as substitution or integration by parts.
Please note that since the integral involves the absolute value function, the area may need to be split into multiple integrals if the curve crosses the x-axis within the given interval.