I really need some help with these Riemann sum problems

[Jimmy84]

Homework Statement

1.
Express as a sum of riemann and write the integral to express the area of the trapezoid with vertex (0,0) , (1,3) , (3,3) , (5,0).

2.
find the intersection points limited by these equations y = xsquare -3x and y = -2x +3 = 0

3.
the trapezoid with vertex (0,0) , (0,2) , (3,2) and (5,0) spins arround the y-axis . Express it as a Riemann sum and write the integral to express the volume of the solid.

Homework Equations

The Attempt at a Solution

My biggest problems in problem 1 is that I don't know how to express the area of the trapezoid as an integral or as a function how can it be?

I have looked for this in many math books but they don't mention this at all.


In problem 2 I need to find the intersection points . I tried seting the two equations equal and then solving for x . but it resulted in a quadratic equation and the results weren't right.

I need help because I ll have a semestral exam tomorrow about this subject and I really need to master it as much as I can.

Thank s a lot in advance for your time.

 

[Jimmy84]

Can someone please help.

I already know how to draw the riemann sum and all the details, I am just looking for a way to describe the trapezoid as a function or as many functions so that I could express that function as a riemann sum.

I would appreciate so much any advice.

 

[Mark44]

Homework Statement

1.
Express as a sum of riemann and write the integral to express the area of the trapezoid with vertex (0,0) , (1,3) , (3,3) , (5,0).

2.
find the intersection points limited by these equations y = xsquare -3x and y = -2x +3 = 0

3.
the trapezoid with vertex (0,0) , (0,2) , (3,2) and (5,0) spins arround the y-axis . Express it as a Riemann sum and write the integral to express the volume of the solid.

Homework Equations

The Attempt at a Solution

My biggest problems in problem 1 is that I don't know how to express the area of the trapezoid as an integral or as a function how can it be?

Break the problem into three parts, with one integral/Riemann sum for the triangular region on the left, another for the rectangular region in the middle, and a third for the triangular region on the right. For the two triangular regions, the upper boundary is part of a straight line, and you know two points on each line, so you should be able to get the equation of each line.

I have looked for this in many math books but they don't mention this at all.


In problem 2 I need to find the intersection points . I tried seting the two equations equal and then solving for x . but it resulted in a quadratic equation and the results weren't right.

What equation did you get? Show us what you did.

I need help because I ll have a semestral exam tomorrow about this subject and I really need to master it as much as I can.

Thank s a lot in advance for your time.

 

[Jimmy84]

Break the problem into three parts, with one integral/Riemann sum for the triangular region on the left, another for the rectangular region in the middle, and a third for the triangular region on the right. For the two triangular regions, the upper boundary is part of a straight line, and you know two points on each line, so you should be able to get the equation of each line.
What equation did you get? Show us what you did.

I think I ve now sovled problem one . in another problem I am given the vertexs (0,0) , (0,2) , (3,3) and (3,0) . in this case the trapezoid has another shape and I can't do what was exactly done in problem one , my guess is that I should divide it in a square and a traingle but for the square should I use an horizontal rectangle to represent its riemann sum?.

In problem 2 i tried to set the two equatiosn equal, then I tried to solve for x, but i wasent able to factorize so i tried to use the quadratic equation to find x in this way

x square -3x = 2x -3 that is , x square -5x +3 this expresion canot be factorized so
by using the quadratic equation resulted in x = (5 +-square root of 13 )/2

this numbers the answer i got was x=4,30 and x=-0,697 .

If I replace those numbers by x in the original equations and don't get the intersects. How can I find them ?



and in problem 3 should I use the same method of dividing the trpezoid in many areas in order to express the volume of the solid?

Thanks.

 

[Mark44]

I think I ve now sovled problem one . in another problem I am given the vertexs (0,0) , (0,2) , (3,3) and (3,0) . in this case the trapezoid has another shape and I can't do what was exactly done in problem one , my guess is that I should divide it in a square and a traingle but for the square should I use an horizontal rectangle to represent its riemann sum?.

In problem 2 i tried to set the two equatiosn equal, then I tried to solve for x, but i wasent able to factorize so i tried to use the quadratic equation to find x in this way

x square -3x = 2x -3 that is , x square -5x +3 this expresion canot be factorized so
by using the quadratic equation resulted in x = (5 +-square root of 13 )/2

These values are fine.

this numbers the answer i got was x=4,30 and x=-0,697 .

If I replace those numbers by x in the original equations and don't get the intersects. How can I find them ?

Use each x value that you got to find the y value at one of the points of intersection. Each intercept has an x-value and a y-value.

and in problem 3 should I use the same method of dividing the trpezoid in many areas in order to express the volume of the solid?

I would use the disk method. After you rotate the trapezoid around the y-axis you get a solid that looks like a cone with the top cut off. Each disk of the solid will have a volume of π * (radius of disk)² * (thickness of disk).

The radius of the disk is the distance from the y-axis to the line that goes through (3, 2) and (5, 0).

 

[Jimmy84]

These values are fine.
Use each x value that you got to find the y value at one of the points of intersection. Each intercept has an x-value and a y-value.
I would use the disk method. After you rotate the trapezoid around the y-axis you get a solid that looks like a cone with the top cut off. Each disk of the solid will have a volume of π * (radius of disk)² * (thickness of disk).

The radius of the disk is the distance from the y-axis to the line that goes through (3, 2) and (5, 0).

For problem 1 my result was

Integral from 0 to 1 of (3x) dx + integral from 1 to 3 of (3) dx + integral from 3 to 5 of ( (-3/2)x + 15/2 ) dx



I was able to find the intersects for problem number 2 .

for problem number 3 the result should be

pi integral from 0 to 3 of (4) dx + pi integral from 3 to 5 of (-x+5)square dx ?




Are the results right ?



-Also for this problem given the vertexs (0,0) , (0,2) , (3,3) and (3,0) find the integrals. in this case the shape of the trapezoid is diferent I used horizontal rectangles to represent riemanns sum.
The result was:

The integral from 0 to 2 of (2) dy + the integral from 2 to 3 of (3y -6) dy

is this correct?

Thanks for your time.