Consider the area shown below. the height of the triangle is 8 and the length of its base is 3. we have used the notation dh for δh.

write a riemann sum for the area, using the strip shown and the variable h: riemann sum =σ now write the integral that gives this area: area =∫ba

The Riemann sum for the area of the triangle is .

The integral that gives the area is , where and

The exact area of the region is 12 square units.

The Riemann sum of the triangle is described by the following formula:

(1)

Where:

Now we derive an expression for the base of the rectangle in terms of the base and height of the triangle:

(2)

Where:

By (2) in (1), we obtain a Riemann sum for the area of the triangle:

(3)

The Riemann sum for the area of the triangle is .

The integral that gives the area of the triangle is based on (3):

The integral that gives the area is , where and .

Now we obtain the exact expression by integration:

If we know that , , and , then the exact area of the region is:

The exact area of the region is 12 square units.

We kindly invite to check this question on area calculations by integrals: link

Step-by-step explanation:

Since we have been given height of triangle as 8 and length of its base as 3. We can use  similar triangles to express the base of the smaller triangle in terms of h.

The height of the smaller triangle will be

Let x be the base of the smaller triangle. Therefore, using similar triangles, we can set the ratios of corresponding sides of the two triangles equal to each other as shown below:

Now, we can express the area of the small rectangular strip of length x and thickness Dh as shown below:

Therefore, the required Riemann sum can be expressed as:

The required areas can be expressed as:

Rest of your answers are correct. :)