Evaluate integral, don't use calculator.
I'll use the S as the sign for integral
S 4 sin (theta)d(theta) from pi/2 to pi
I know the answer is 4 because I enter it in on my calculator, but don't know the steps to get to the answer.
Please help.
The steps are:
(1) compute the indefinite integral.
in this case it is -4 cos theta
(2) Compute the value of the indefinite integral at theta = pi and subtract the value at theta = pi/2.
That would be -4(-1) -0 = 4
You won't learn integral calculus without learning how to compute integrals.
To evaluate the integral ∫ 4 sin (θ) dθ from π/2 to π without using a calculator, you can use the properties and rules of integrals.
Step 1: Find the antiderivative of sin(θ).
The antiderivative of sin(θ) is -cos(θ). So, the integral becomes:
∫ 4 sin (θ) dθ = -4 cos(θ) + C
Step 2: Evaluate the integral.
To evaluate the definite integral from π/2 to π, substitute the upper limit (π) and lower limit (π/2) into the antiderivative:
[-4 cos(θ)] [from π/2 to π]
Evaluate the upper limit:
Plugging in π into the antiderivative gives:
[-4 cos(π)]
Cosine of π is -1, so the expression becomes:
[-4 * (-1)] = 4
Evaluate the lower limit:
Plugging in π/2 into the antiderivative gives:
[-4 cos(π/2)]
Cosine of π/2 is 0, so the expression becomes:
[-4 * 0] = 0
Step 3: Subtract the lower limit from the upper limit:
Subtracting the lower limit (0) from the upper limit (4) gives the final answer:
4 - 0 = 4
So, the value of the integral ∫ 4 sin (θ) dθ from π/2 to π is 4.