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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

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 ar class=

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Answer:

[tex]\text{Riemann sum }=\sum \frac{3}{8}(8-h)Dh[/tex]

[tex]\text{Area =}\int_{a}^{b} \frac{3}{8}(8-h)Dh[/tex]

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 [tex](8-h)[/tex]

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:

[tex]\frac{8-h}{x} =\frac{8}{3} \\x=\frac{3}{8}(8-h)[/tex]

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

[tex]DA=x*Dh\\DA=\frac{3}{8}(8-h)Dh[/tex]

Therefore, the required Riemann sum can be expressed as:

[tex]\text{Riemann sum }=\sum \frac{3}{8}(8-h)Dh[/tex]

The required areas can be expressed as:

[tex]\text{Area =}\int_{a}^{b} \frac{3}{8}(8-h)Dh[/tex]

Rest of your answers are correct. :)

xero099

The Riemann sum for the area of the triangle is [tex]A = B\cdot \Sigma \left(1-\frac{h}{H} \right) \,\Delta h[/tex].

The integral that gives the area is [tex]A = B\left[ \int\limits^H_0\, dh -\frac{1}{H} \int\limits^H_0 {h} \, dh\right][/tex], where [tex]a = 0[/tex] and [tex]b = H[/tex]

The exact area of the region is 12 square units.

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

[tex]A = \Sigma\, b(h) \Delta h[/tex] (1)

Where:

  • [tex]b(h)[/tex] - Base of the rectangle.
  • [tex]\Delta h[/tex] - Height of the rectangle.
  • [tex]A[/tex] - Area of the rectangle.

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

[tex]\frac{b(h)}{B} = \frac{H-h}{H}[/tex]

[tex]\frac{b(h)}{B} = 1 -\frac{h}{H}[/tex] (2)

Where:

  • [tex]B[/tex] - Base of the triangle.
  • [tex]H[/tex] - Height of the triangle.
  • [tex]h[/tex] - Position of the rectangle within the triangle.

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

[tex]A = B\cdot \Sigma \left(1-\frac{h}{H} \right) \,\Delta h[/tex] (3)

The Riemann sum for the area of the triangle is [tex]A = B\cdot \Sigma \left(1-\frac{h}{H} \right) \,\Delta h[/tex].

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

[tex]A = B\left[ \int\limits^H_0\, dh -\frac{1}{H} \int\limits^H_0 {h} \, dh\right][/tex]

The integral that gives the area is [tex]A = B\left[ \int\limits^H_0\, dh -\frac{1}{H} \int\limits^H_0 {h} \, dh\right][/tex], where [tex]a = 0[/tex] and [tex]b = H[/tex].

Now we obtain the exact expression by integration:

[tex]A = B\cdot \left[(H-0)-\frac{1}{2\cdot H}\cdot (H^{2}-0^{2}) \right][/tex]

[tex]A = \frac{1}{2}\cdot B\cdot H[/tex]

If we know that [tex]B = 3[/tex], [tex]H = 8[/tex], [tex]a = 0[/tex] and [tex]b = H[/tex], then the exact area of the region is:

[tex]A = \frac{1}{2}\cdot (3)\cdot (8)[/tex]

[tex]A = 12[/tex]

The exact area of the region is 12 square units.

We kindly invite to check this question on area calculations by integrals: https://brainly.com/question/25412968

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