Answer :
To approximate the area between the [tex]$x$[/tex]-axis and the function [tex]\( g(x) \)[/tex] from [tex]\( x = 10 \)[/tex] to [tex]\( x = 16 \)[/tex] using a left Riemann sum with 3 unequal subdivisions, follow these steps:
1. Identify the subintervals:
- The given [tex]\( x \)[/tex]-values delineate the subintervals: [tex]\( [10, 12] \)[/tex], [tex]\( [12, 15] \)[/tex], and [tex]\( [15, 16] \)[/tex].
2. Determine the width of each subinterval:
- The width of the first subinterval [tex]\( [10, 12] \)[/tex] is [tex]\( 12 - 10 = 2 \)[/tex].
- The width of the second subinterval [tex]\( [12, 15] \)[/tex] is [tex]\( 15 - 12 = 3 \)[/tex].
- The width of the third subinterval [tex]\( [15, 16] \)[/tex] is [tex]\( 16 - 15 = 1 \)[/tex].
3. Use the left endpoint of each subinterval for the function value:
- For [tex]\( [10, 12] \)[/tex], use [tex]\( g(10) = 5 \)[/tex].
- For [tex]\( [12, 15] \)[/tex], use [tex]\( g(12) = 1 \)[/tex].
- For [tex]\( [15, 16] \)[/tex], use [tex]\( g(15) = 7 \)[/tex].
4. Calculate the area of each rectangle:
- The area for the first subinterval is [tex]\( width \times height = 2 \times 5 = 10 \)[/tex].
- The area for the second subinterval is [tex]\( width \times height = 3 \times 1 = 3 \)[/tex].
- The area for the third subinterval is [tex]\( width \times height = 1 \times 7 = 7 \)[/tex].
5. Sum the areas of the rectangles:
- Total approximate area = [tex]\( 10 + 3 + 7 = 20 \)[/tex].
Thus, the approximate area between the [tex]$x$[/tex]-axis and [tex]\( g(x) \)[/tex] from [tex]\( x = 10 \)[/tex] to [tex]\( x = 16 \)[/tex] is 20 square units.
1. Identify the subintervals:
- The given [tex]\( x \)[/tex]-values delineate the subintervals: [tex]\( [10, 12] \)[/tex], [tex]\( [12, 15] \)[/tex], and [tex]\( [15, 16] \)[/tex].
2. Determine the width of each subinterval:
- The width of the first subinterval [tex]\( [10, 12] \)[/tex] is [tex]\( 12 - 10 = 2 \)[/tex].
- The width of the second subinterval [tex]\( [12, 15] \)[/tex] is [tex]\( 15 - 12 = 3 \)[/tex].
- The width of the third subinterval [tex]\( [15, 16] \)[/tex] is [tex]\( 16 - 15 = 1 \)[/tex].
3. Use the left endpoint of each subinterval for the function value:
- For [tex]\( [10, 12] \)[/tex], use [tex]\( g(10) = 5 \)[/tex].
- For [tex]\( [12, 15] \)[/tex], use [tex]\( g(12) = 1 \)[/tex].
- For [tex]\( [15, 16] \)[/tex], use [tex]\( g(15) = 7 \)[/tex].
4. Calculate the area of each rectangle:
- The area for the first subinterval is [tex]\( width \times height = 2 \times 5 = 10 \)[/tex].
- The area for the second subinterval is [tex]\( width \times height = 3 \times 1 = 3 \)[/tex].
- The area for the third subinterval is [tex]\( width \times height = 1 \times 7 = 7 \)[/tex].
5. Sum the areas of the rectangles:
- Total approximate area = [tex]\( 10 + 3 + 7 = 20 \)[/tex].
Thus, the approximate area between the [tex]$x$[/tex]-axis and [tex]\( g(x) \)[/tex] from [tex]\( x = 10 \)[/tex] to [tex]\( x = 16 \)[/tex] is 20 square units.
