Answer :
To approximate the area between the [tex]$x$[/tex]-axis and [tex]$g(x)$[/tex] from [tex]$x=10$[/tex] to [tex]$x=16$[/tex] using a left Riemann sum with 3 subdivisions, follow these steps:
1. Identify the subdivision points:
[tex]$$x = 10, \quad 12, \quad 15, \quad 16.$$[/tex]
Thus, the subintervals are:
- From [tex]$10$[/tex] to [tex]$12$[/tex]
- From [tex]$12$[/tex] to [tex]$15$[/tex]
- From [tex]$15$[/tex] to [tex]$16$[/tex]
2. Compute the width of each subinterval:
- For the first subinterval:
[tex]$$\Delta x_1 = 12 - 10 = 2.$$[/tex]
- For the second subinterval:
[tex]$$\Delta x_2 = 15 - 12 = 3.$$[/tex]
- For the third subinterval:
[tex]$$\Delta x_3 = 16 - 15 = 1.$$[/tex]
3. Use the left endpoint values to compute the area of each subinterval:
- For the first subinterval, the left endpoint is at [tex]$x=10$[/tex] with [tex]$g(10)=5$[/tex], so the area is:
[tex]$$\text{Area}_1 = g(10) \cdot \Delta x_1 = 5 \cdot 2 = 10.$$[/tex]
- For the second subinterval, the left endpoint is at [tex]$x=12$[/tex] with [tex]$g(12)=1$[/tex], so the area is:
[tex]$$\text{Area}_2 = g(12) \cdot \Delta x_2 = 1 \cdot 3 = 3.$$[/tex]
- For the third subinterval, the left endpoint is at [tex]$x=15$[/tex] with [tex]$g(15)=7$[/tex], so the area is:
[tex]$$\text{Area}_3 = g(15) \cdot \Delta x_3 = 7 \cdot 1 = 7.$$[/tex]
4. Sum the areas of all subintervals to obtain the approximate total area:
[tex]$$\text{Total Area} = \text{Area}_1 + \text{Area}_2 + \text{Area}_3 = 10 + 3 + 7 = 20.$$[/tex]
Thus, the approximate area between the [tex]$x$[/tex]-axis and [tex]$g(x)$[/tex] on the given interval is
[tex]$$20 \text{ units}^2.$$[/tex]
1. Identify the subdivision points:
[tex]$$x = 10, \quad 12, \quad 15, \quad 16.$$[/tex]
Thus, the subintervals are:
- From [tex]$10$[/tex] to [tex]$12$[/tex]
- From [tex]$12$[/tex] to [tex]$15$[/tex]
- From [tex]$15$[/tex] to [tex]$16$[/tex]
2. Compute the width of each subinterval:
- For the first subinterval:
[tex]$$\Delta x_1 = 12 - 10 = 2.$$[/tex]
- For the second subinterval:
[tex]$$\Delta x_2 = 15 - 12 = 3.$$[/tex]
- For the third subinterval:
[tex]$$\Delta x_3 = 16 - 15 = 1.$$[/tex]
3. Use the left endpoint values to compute the area of each subinterval:
- For the first subinterval, the left endpoint is at [tex]$x=10$[/tex] with [tex]$g(10)=5$[/tex], so the area is:
[tex]$$\text{Area}_1 = g(10) \cdot \Delta x_1 = 5 \cdot 2 = 10.$$[/tex]
- For the second subinterval, the left endpoint is at [tex]$x=12$[/tex] with [tex]$g(12)=1$[/tex], so the area is:
[tex]$$\text{Area}_2 = g(12) \cdot \Delta x_2 = 1 \cdot 3 = 3.$$[/tex]
- For the third subinterval, the left endpoint is at [tex]$x=15$[/tex] with [tex]$g(15)=7$[/tex], so the area is:
[tex]$$\text{Area}_3 = g(15) \cdot \Delta x_3 = 7 \cdot 1 = 7.$$[/tex]
4. Sum the areas of all subintervals to obtain the approximate total area:
[tex]$$\text{Total Area} = \text{Area}_1 + \text{Area}_2 + \text{Area}_3 = 10 + 3 + 7 = 20.$$[/tex]
Thus, the approximate area between the [tex]$x$[/tex]-axis and [tex]$g(x)$[/tex] on the given interval is
[tex]$$20 \text{ units}^2.$$[/tex]
