Approximate the definite integral by dividing the interval into 4 subintervals and using the left endpoint of each subinterval.

Question

a. 3432

b. 2766

c.2765

d. 2152

e. 2144

f. none of the above

Really confused about this question. Thank you for the help in advance.

Answer 1

What you need to use here is the Riemann Sum from the left-end point. The formula is:

∫^b_af(x)dx = Δx•(f(x₀)+f(x₁)+f(x₂)+...+f(x_n-1)) where Δx = (b-a)/n

So for: ∫¹²₄ (5x² - 1)dx, n = 4

First, solve for Δx:

Δx = (12-4)/4 = 8/4 = 2

This means that we have to get the 4 numbers for x₀...x_n-1 2 units apart starting from a=4 to something less than 12. (Note: It's left-endpoint, that's why 12 is not included)

x₀ = 4

x₁ = 6

x₂ = 8

x₃ = 10

Therefore:

∫¹²₄ (5x² - 1)dx = 2 (f(4) + f(6) + f(8) + f(10))

f(4) = 5•4² - 1 = 79

f(6) = 5•6² - 1 = 179

f(8) = 5•8² - 1 = 319

f(10) = 5•10² - 1 = 499

∫¹²₄ (5x² - 1)dx = 2 (79+ 179 + 319 + 499) = 2152 sq. units

Answer 2

So Viola S.,

the problem is trying to get you to just calculate that function at four points, namely, at x = 4, 6, 8, and 10. Then you generate little horizontal boxes out to the next x value, and add the area values of those 4 little stepped boxes. Note that this is a "stairstepped" approximation of the exact integral. If you made the steps ("subintervals") very small in width, you approach the exact integral better and better. Once you perform the integration, you may calculate the area under the function (between x=4, x=12, the function curve, and the x-axis) exactly.

-- Cheers, --Mr. d.