Question

Approximate ∫14x3dx Using A Left Reimann Sum With Partition Points At {1,1.5, And 2.5}.

A 1³(1)+1.5³(1.5)+2.5³(1)+4³(1.5)
B 1³(0.5)+1.5³(1)+2.5³(1.5)
C 1³(1)+1.5³(1)+2.5³(1)+43(1)
D 1³ (1)+1.5³(1)+2.5³(1)

Answer 1

The approximation of the integral ∫14x3dx using a Left Riemann Sum with partition points at {1, 1.5, and 2.5} is obtained by evaluating the function at each left endpoint and multiplying by the interval width, which results in 3.875, corresponding to choice B without considering the 2.5 term.

Step-by-step explanation:

To approximate the integral ∫14x3dx using a Left Riemann Sum with partition points at {1, 1.5, and 2.5}, we first evaluate the function at each left endpoint of the partitions and then multiply each value by the width of the subinterval to find the area of rectangles that approximate the area under the curve. Since the width between 1 and 1.5 is 0.5, and between 1.5 and 2.5 is 1, we only consider the left endpoints (1 and 1.5) for our sum:

• The value at x = 1 is 13 = 1
• The value at x = 1.5 is (1.5)3 = 3.375

The approximate sum is then 13(0.5) + 1.53(1). Therefore, after performing these multiplications, we get an approximation of:

(1)(0.5) + (3.375)(1) = 0.5 + 3.375 = 3.875.

The correct choice that represents this calculation is answer B, which states 13(0.5) + 1.53(1) + 2.53(1.5). However, since there is no subinterval from 1.5 to 2 (the width here should be 0.5, not 1.5), we can disregard 2.53(1.5) as part of our answer, leaving us with the Left Riemann Sum for the intervals given as 3.875.