Approximate the area under the curve over the specified interval by using the indicated number of subintervals (or rectangles) and evaluating the function at the left-hand endpoints of the subintervals. Compute the approximate area using up to 4 decimal places as needed. from to subintervals.

A)11.625 square units
B)23.25 square units
C)19.25 square units
D)46.5 square units
E)2 square units

Question

Approximate the area under the curve over the specified interval by using the indicated number of subintervals (or rectangles) and evaluating the function at the left-hand endpoints of the subintervals. Compute the approximate area using up to 4 decimal places as needed. ​  from  to   subintervals.
​
A)11.625 square units 
B)23.25 square units 
C)19.25 square units 
D)46.5 square units 
E)2 square units

Quizplus · Accepted Answer

Using the left-hand endpoint method with 4 subintervals, we have:
$\Delta x=\frac{b-a}{n}=\frac{5-1}{4}=1$
$x_0=1, x_1=2, x_2=3, x_3=4, x_4=5$
$f(x_0)=f(1)=1,f(x_1)=f(2)=3,f(x_2)=f(3)=5,f(x_3)=f(4)=7$
$	ext{Area}\approx\sum_{i=0}^{n-1}f(x_i)\Delta x = f(x_0)\Delta x+f(x_1)\Delta x+f(x_2)\Delta x+f(x_3)\Delta x$
$=1(1)+3(1)+5(1)+7(1)=16$
Using the left-hand endpoint method with 8 subintervals, we have:
$\Delta x=\frac{b-a}{n}=\frac{5-1}{8}=0.5$
$x_0=1, x_1=1.5, x_2=2, x_3=2.5, x_4=3, x_5=3.5, x_6=4,x_7=4.5,x_8=5$
$f(x_0)=f(1)=1, f(x_1)=f(1.5)=1.5, f(x_2)=f(2)=3, f(x_3)=f(2.5)=3.5, f(x_4)=f(3)=5, f(x_5)=f(3.5)=5.5$
$f(x_6)=f(4)=7, f(x_7)=f(4.5)=7.5$
$	ext{Area}\approx\sum_{i=0}^{n-1}f(x_i)\Delta x=f(x_0)\Delta x+f(x_1)\Delta x+f(x_2)\Delta x+f(x_3)\Delta x$
$+f(x_4)\Delta x+f(x_5)\Delta x+f(x_6)\Delta x+f(x_7)\Delta x=1(0.5)+1.5(0.5)+3(0.5)+3.5(0.5)+5(0.5)+5.5(0.5)+7(0.5)+7.5(0.5)=23.25$
Therefore, the best choice is B, as it gives a more accurate approximation than using 4 subintervals.

Approximate the Area Under the Curve Over the Specified Interval