find the area enclosed by y=x^2 and y=sinx in the first quadrant.
The hardest part is to find the intersection of the two curves.
I don't know what level of Calculus this is but the equation is nasty to solve.
WolfFram gave me
http://www.wolframalpha.com/input/?i=x%5E2+%3D+sin%28x%29
x = ..876726 and of course the obvious x = 0
so area = ∫(sinx - x^2) dx from 0 to .876726
= [ -cosx - x^3/3 ]
= -.63967 - .22463 - (-1 - 0)
= .1356975
x^2 = sinx at x=.876
So integrate
sin(x)-x^2 from 0 to .876
To find the area enclosed by the curves y = x^2 and y = sin(x) in the first quadrant, we need to find the points of intersection and then integrate the difference of the y-values over that interval.
Step 1: Find the points of intersection:
To find the points where the two curves intersect, we need to solve the equation x^2 = sin(x) in the first quadrant. Unfortunately, this equation cannot be solved algebraically. We need to use numerical methods or graphing calculator.
Using a graphing calculator or a computer software, plot the graphs of y = x^2 and y = sin(x). You will see that they intersect at two points in the first quadrant. Let's denote these points as P and Q.
Step 2: Determine the limits of integration:
We need to find the x-values of the points of intersection. Let's denote the x-coordinate of point P as a and the x-coordinate of point Q as b. These are the limits of integration for calculating the area.
Step 3: Set up the integral:
The area enclosed by the curves y = x^2 and y = sin(x) in the first quadrant can be found by integrating the difference of the y-values, which is (x^2 - sin(x)), between the limits a and b:
Area = ∫(a to b) (x^2 - sin(x)) dx
Step 4: Evaluate the integral:
Evaluate the integral using appropriate techniques, such as integration by parts or numerical integration methods. Depending on the accuracy required, you can use numerical approximations like Riemann sums or numerical integration methods like Simpson's rule or the trapezoidal rule.
Once you have the integral evaluated, you will have the area enclosed by the curves y = x^2 and y = sin(x) in the first quadrant.