How many definite integrals would be required to represent the area of the region enclosed by the curves y=(cos^2(x))(sin(x)) and y=0.03x^2, assuming you could not use the absolute value function?
a.) 1
b.) 2
c.) 3
d.) 4
e.) 5
3 I believe
Just look at the graph
Well, I guess you could say the number of definite integrals required to represent the area of the region enclosed by those curves is... "integral-ly" dependent on a few factors. But fear not, I shall simplify it for you!
Let's take a look at the two curves involved. The first curve is y = (cos^2(x))(sin(x)). The second curve is y = 0.03x^2. To find the area enclosed by these curves, we need to find the points of intersection first.
Now, it so happens that there are four points of intersection between these two curves. That means the answer to our question is... (drumroll please) option d.) 4!
So, you'll need a total of four definite integrals to represent the area of the region enclosed by these curves. Of course, you might be tempted to shout "eureka!" after the first integral, but the other three are waiting for you. Keep on integrating, my friend!
To find the number of definite integrals required to represent the area of the region enclosed by the given curves, we need to find the points of intersection of the curves. The region enclosed by the curves is bounded by the curves themselves and the x-axis.
First, let's set the two curves equal to each other and solve for x to find the points of intersection:
(cos^2(x))(sin(x)) = 0.03x^2
To solve this equation algebraically, we can use a graphing calculator or a computer program. However, since we cannot use the absolute value function, we need to break down the equation into separate cases.
Case 1: cos^2(x) = 0 and sin(x) ≠ 0
From the equation cos^2(x) = 0, we know that cos(x) = 0. Therefore, we find the first solution x = π/2.
Case 2: cos^2(x) ≠ 0 and sin(x) = 0
Since sin(x) = 0, we know that x = kπ, where k is an integer. However, in this case, it cannot be a solution, as it does not satisfy the equation (cos^2(x))(sin(x)) = 0.03x^2.
Case 3: cos^2(x) = 0 and sin(x) = 0
This case does not produce any solutions to the given equation.
Now, we can draw the graphs of the two curves on an x-y plane and identify the regions between the curves and the x-axis. To obtain this graph, we can input the equations into a graphing calculator or a computer program.
Upon inspecting the graphs, we can see that there are three regions enclosed by the curves and the x-axis. The first region is between x = 0 and x = π/2, the second region is between x = π/2 and x = 2π, and the third region is between x = 2π and x = 3π/2.
Since there are three enclosed regions, we would require three definite integrals to represent the area of the region enclosed by the curves.
Therefore, the answer is c.) 3.
See plot and make your choice.
http://img401.imageshack.us/img401/2217/1330563319.png