Follow the Fundamental Theorem of Calculus.
PROBLEM 6 . Given the following table of values for t and f(t):
Use a right-hand Riemann sum with 6 subintervals indicated by the data in the table to approximate
f(t) dt.
Answer: The width of the first rectangle is 2 − 0 = 2; the second width is 4 − 2 = 2; the third is 7 − 4 = 3;
the fourth is 11 − 7 = 4; the fifth is 13 − 11 = 2; and the sixth is 14 − 13 = 1. We find the height of each
rectangle by evaluating f(t) at the appropriate value of t, the right endpoint of each interval on the t-axis.
Here y 1 = 6, y 2 = 10, y 3 = 15 y 4 = 20, y 5 = 26, and y 6 = 30. Therefore, we can approximate the integral
with
f(t) dt = (2)(6) + (2)(10) + (3)(15) + (4)(20) + (2)(26) + (1)(30) = 239PRACTICE PROBLEM SET 20
Here’s a great opportunity to practice finding the area beneath a curve and evaluating integrals. The
answers are in Chapter 19.
1.Find the area under the curve y = 2x − x^2 from x = 1 to x = 2 with n = 4 left-endpoint rectangles.2.Find the area under the curve y = 2x − x^2 from x = 1 to x = 2 with n = 4 right-endpoint rectangles.3.Find the area under the curve y = 2x − x^2 from x = 1 to x = 2 using the Trapezoid Rule with n = 4.