Find the area of the region IN THE FIRST QUADRANT (upper right quadrant) bounded by the curves y=sin(x)cos(x)^2, y=2xcos(x^2) and y=4-4x.
You get:
a.)1.8467
b.) 0.16165
c.) 0.36974
d.) 1.7281
e.) 0.37859
Based on my calculations, I would say that the answer is e.) 0.37859. I am checking my answer.
That's the correct answer.
If you integrated between the end-points 0, 0.69275, 0.92811, you should have got the areas 0.28024 and 0.09835 respectively, which add up to 0.37859.
Thank you, I was just really unsure of my answer!
You're welcome! :)
To find the area of the region bounded by the curves y = sin(x)cos(x)^2, y = 2xcos(x^2), and y = 4 - 4x in the first quadrant, you can use the concept of definite integrals.
First, we need to find the x-values where the curves intersect. To do this, we set the equations equal to each other and solve for x.
sin(x)cos(x)^2 = 2xcos(x^2) ---> sin(x)cos(x)^2 - 2xcos(x^2) = 0
Unfortunately, finding the exact x-values of the intersections is quite difficult because it involves solving a transcendental equation. So we will rely on numerical methods or a graphing calculator to approximate the intersection points.
After finding the approximate intersection points, we need to determine the limits of integration for each curve. Since we are only interested in the first quadrant, we only need to consider the positive x-values of the intersections.
Once we have the interval of integration, calculate the area between the curves by taking the definite integral of the difference between the upper curve and the lower curve with respect to x over the given interval.
The formula for the area between two curves is:
Area = ∫ (upper curve - lower curve) dx
After evaluating the definite integral, you should obtain the approximate area.
Based on your calculations, your answer is e.) 0.37859. If you are checking your answer, well done! Make sure to verify the accuracy of your calculations and double-check the limits of integration before finalizing your answer.