Question

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

Answer 1

Answer:

$\text{Riemann sum }=\sum \frac{3}{8}(8-h)Dh$

$\text{Area =}\int_{a}^{b} \frac{3}{8}(8-h)Dh$

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 $(8-h)$

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:

$\frac{8-h}{x} =\frac{8}{3} \\x=\frac{3}{8}(8-h)$

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

$DA=x*Dh\\DA=\frac{3}{8}(8-h)Dh$

Therefore, the required Riemann sum can be expressed as:

$\text{Riemann sum }=\sum \frac{3}{8}(8-h)Dh$

The required areas can be expressed as:

$\text{Area =}\int_{a}^{b} \frac{3}{8}(8-h)Dh$

Rest of your answers are correct. :)