Find its area. y = sec^2(x), y = 8 cos(x), −π/3 ≤ x ≤ π/3
For sinπ/3, I got 16√(3)/2 but for tanπ/3, I don't know how to find the answer for it. I know at tanπ/3, sin=3/√(2) and cos= 1/√(2). Should that give me √(3)/2? So I got 16√(3)/2 -2√(3)/√(2) is the wrong answer
tan π/3 = √3
tan π/6 = 1/√3
One of those "standard" values.
learn it, love it.
like pi/3 = 60 degrees and pi/6 = 30 degrees :)
To find the area between two curves, you need to first find the points of intersection between the curves. In this case, the two curves are y = sec^2(x) and y = 8 cos(x).
To find the points of intersection, set the two equations equal to each other:
sec^2(x) = 8 cos(x)
Now, we can solve this equation for x.
Dividing both sides by cos^2(x):
1 = 8 sin^2(x)
Rearranging the equation:
sin^2(x) = 1/8
Taking the square root of both sides:
sin(x) = ±√(1/8)
Now, we need to find the values of x that satisfy this equation. Since -π/3 ≤ x ≤ π/3, we only need to find the value of x within this range.
Let's solve for sin(x) = √(1/8):
x = arcsin(√(1/8))
Using a calculator, we find that x ≈ 0.616.
Now, let's solve for sin(x) = -√(1/8):
x = arcsin(-√(1/8))
Using a calculator, we find that x ≈ -0.616.
So, the points of intersection are approximately x = 0.616 and x = -0.616.
To find the area between the curves, we need to integrate the difference of the two curves with respect to x over the interval -π/3 ≤ x ≤ π/3.
The formula to find the area between two curves is:
Area = ∫[a, b] (f(x) - g(x)) dx
where f(x) and g(x) are the two curves, and a and b are the x-coordinates of the points of intersection.
In this case, the area between the curves is:
Area = ∫[-0.616, 0.616] (sec^2(x) - 8cos(x)) dx
Evaluate this definite integral to find the area.
Let me know what you get, they probably want an "exact" answer.
Notice that at the intersection of the two curves, x = π/3
and we have symmetry, so we can take the area from 0 to π/3, then double
area = 2∫ (8cosx - sec^2 x) dx from 0 to π/3
= 2[8sinx - tanx] from 0 to π/3
= 2(8sin π/3 - tan π/3 - (8sin 0 - tan 0 )
= ......
your turn