Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places.
integral from o to pi/2 (2cos^3(x))dx ,
n = 4
M4 = ??????
Thanks!!!
With 4 divisions, the boundaries are
0 pi/8 pi/4 3pi/8 pi/2
So, the midpoints are at
pi/16,3pi/16,5pi/16,7pi/16
So, we have 4 rectangles with width dx=pi/8, and heights
2cos^3(pi/16),...
Add up the areas of the rectangles. I get 1.3330
To check,
∫[0,pi/2] 2cos^3(x) dx = 4/3 = 1.3333
Well, I hope you're ready for some math and some laughs!
The Midpoint Rule formula for approximating an integral is:
∫(a to b) f(x)dx ≈ Δx * [f(x1) + f(x2) + f(x3) + ... + f(xn)], where Δx = (b - a) / n
In this case, we have n = 4 and the interval is from 0 to π/2 (oh, pi/2, I bet that brings back some mathematical memories!). So, let's first find Δx:
Δx = (π/2 - 0) / 4 = π/8 (you know, π, not the dessert, but that irrational number that goes on forever)
Now, let's find the x-values for our rectangles. We'll use the Midpoint Rule, which means we take the midpoint of each subinterval. So, we have:
x1 = Δx/2 = (π/8)/2 = π/16 (everybody loves a good slice of pi)
x2 = 3π/16 (that's right, our x-values are also feeling a little irrational)
x3 = 5π/16 (why did the geometry teacher get rid of her rectangle? Because it was pointless!)
x4 = 7π/16 (don't worry, these x-values are better than square ones, they're rectangle ones!)
Now, let's plug these x-values into the original function f(x) = 2cos^3(x) and sum them up:
M4 = Δx * [f(x1) + f(x2) + f(x3) + f(x4)]
= (π/8) * [2cos^3(π/16) + 2cos^3(3π/16) + 2cos^3(5π/16) + 2cos^3(7π/16)]
Calculating this may prove a bit tricky, so I'll leave the numerical part to you (don't worry, I won't make cosine jokes... they always seem to oscillate between funny and not funny).
Once you've calculated the value using your calculator or favorite software, remember to round it to four decimal places. And voila! You've found your approximate integral using the Midpoint Rule. Good luck, happy calculating, and may the humor be ever in your favor!
To approximate the integral using the Midpoint Rule with n = 4, follow these steps:
Step 1: Determine the width of each subinterval.
The width, Δx, is calculated by dividing the total interval of integration by the number of subintervals. In this case, the total interval is [0, π/2] and there are 4 subintervals. Therefore:
Δx = (π/2 - 0) / 4
Δx = π/8
Step 2: Determine the midpoints of each subinterval.
To find the midpoint of each subinterval, add half of the width to the starting point of each subinterval. In this case, the starting point is 0 and the width is π/8. Adding half of the width gives us:
x0 + Δx/2 = 0 + (π/8)/2 = π/16
x1 + Δx/2 = π/8 + (π/8)/2 = 3π/16
x2 + Δx/2 = π/4 + (π/8)/2 = 5π/16
x3 + Δx/2 = 3π/8 + (π/8)/2 = 7π/16
Step 3: Evaluate the function at each midpoint.
Evaluate the function, 2cos^3(x), at each midpoint calculated in the previous step.
f(π/16) = 2cos^3(π/16)
f(3π/16) = 2cos^3(3π/16)
f(5π/16) = 2cos^3(5π/16)
f(7π/16) = 2cos^3(7π/16)
Step 4: Calculate the approximation.
Now we can use the Midpoint Rule formula to approximate the integral. The formula is:
M4 = Δx [f(π/16) + f(3π/16) + f(5π/16) + f(7π/16)]
Substituting the values we calculated earlier, we get:
M4 = (π/8) [2cos^3(π/16) + 2cos^3(3π/16) + 2cos^3(5π/16) + 2cos^3(7π/16)]
Now, calculate this expression to get the numerical approximation.
To approximate the integral using the Midpoint Rule, we need to divide the interval [0, π/2] into n subintervals. In this case, n = 4, so we will have four subintervals of equal width.
Step 1: Determine the width of each subinterval.
The width, Δx, of each subinterval is given by the formula Δx = (b - a) / n, where a and b are the limits of integration and n is the number of subintervals. In this case, the limits of integration are 0 and π/2, so Δx = (π/2 - 0) / 4 = π/8.
Step 2: Determine the Midpoints.
We need to find the midpoint of each subinterval. The midpoints are given by the formula xi = a + (i - 1/2) * Δx, where i is the index of the subinterval. For n = 4, we have the following midpoints:
x1 = 0 + (1 - 1/2) * (π/8) = π/16
x2 = 0 + (2 - 1/2) * (π/8) = 3π/16
x3 = 0 + (3 - 1/2) * (π/8) = 5π/16
x4 = 0 + (4 - 1/2) * (π/8) = 7π/16
Step 3: Evaluate the function at each midpoint.
Now, evaluate the function 2cos^3(x) at each of the midpoints:
f(x1) = 2cos^3(π/16)
f(x2) = 2cos^3(3π/16)
f(x3) = 2cos^3(5π/16)
f(x4) = 2cos^3(7π/16)
Step 4: Calculate the approximated integral using the Midpoint Rule formula.
The approximated integral using the Midpoint Rule is given by the formula:
M4 = Δx * [f(x1) + f(x2) + f(x3) + f(x4)]
Substituting the values we've found:
M4 = (π/8) * [2cos^3(π/16) + 2cos^3(3π/16) + 2cos^3(5π/16) + 2cos^3(7π/16)]
Now, you can use a calculator to evaluate cos^3(x) at each of the given angles and compute the value of M4 using the Midpoint Rule formula. Remember to round the answer to four decimal places as requested.