I need help with this calculus problem. I would like to know if I have worked it correctly and would like to know the correct answer and how to get it if I have done it wrong. Thanks in advance to anyone who can help.

Here is the problem:

In this exercise, the region S is viewed as a type 2 region.

Let S be the region bounded on the left and on the right by the graphs of x=(y^2)+ 2y - 1 and x= -2y +4. Note that the x-axis and y-axis are not bounding curves in this problem.

a.) Determine the intersection points of the above curves. You must list these two points as ordered pairs in the form (x,y). Explicitly show your hand calculations to obtain these points.

My answer: (y^2)+2y-1=-2y +4
(y^2)+4y-5=0
(y+5)(y-1)=0
Y=-5, y=1

Y=-5 x=g(-5)= -2(-5) + 4
= 10 + 4
= 14

Y=1 x= g(1)= -2(1) + 4
= -2 + 4
= 2

Points are (14,-5) and (2,1)

b.) Determine the vertex of the parabola. You must list this as an ordered pair in the form (x,y). Explicitly show your hand calculation to obtain this point.

My answer: f(y)= 2y+2
2y+2=0
2y=-2
2y/2=-2/2
y=-1

x=f(-1)=1+-2-1
=-2
Vertex: (-1,-2)

c.) Sketch the region S. Label the graphs that define the boundary of S. Determine the interval [c,d] on the y-axis over which the region S is defined and label the values c and d on your sketch.

My answer: I am not really sure how to do this part, but if theres anyone who could explain to me how to do it I would greatly appreciate it.

d.) In this part, you are asked to express symbolically the area A of the region S as a limit of a sum. Assume that n rectangles, where these rectangles are of equal width and the heights are determined by the upper endpoints of the subintervals of [c.d] are used to approximate A.

d1.) What is the width of each rectangle? Your answer should be in terms of n only. Note that width refers to delta y in this problem Also find a forumla y sub i in terms of i and n only.

My answer:

Delta y= (b-a)/(n)= (-5-1)/(n)
Delta y =(-6)/n

y sub I= a + i delta y= 1+i (-6/n)
y sub 1= 1+ (-6i)/n

d2.) What is the formula for the height of the ith rectangle? Your answer should be an algebraic expression in terms of n and i only. Do not simplify your answer . Your final answer must not contain any other letters except for n and i.

My answer: hi= f(xi) -g(xi)
=yi^2 + 2yi - 1 -(-2yi+4)
=yi^2 + 4yi - 5
height =(1+ (-6i)/n)^2 + 4 (1+ (-6i)/n)-5

d3). What is the area of the ith rectangle? Your answer should be in terms of n and i only. Do not simplify your answer.

My answer:
Ai= hi * delta y
=((1+(6i)/n)^2 + 4(1+(-6i/n)) -5) * (5/n)

D4.) What is the symbolic expression for the appropriate value, in this case, of the area of the region S? Use your answer to parts d1, d3 and the sigma sign.

My answer: I am not really sure how to do this. Im guessing I write the sigma sign with an n above it and
i=1 below it and then just write my answer from d3 beside it. If anyone could explain to me how to do this I would greatly appreciate it.

d5.) What is the symbolic expression for the exact value of the area of the region S, expressed as a limit of a sum containing only the variables n and i and not containg the letters f,a,b,c,d,x, or g.

Not really sure how to do this part either, and would appreciate any help.

e.) Find the exact numeric value of the area of the region S using integration. Provide sufficient work to justify your numeric answer including an explicit ant derivative with the calculations using the Fundaemental Theorem of Calculus.


I am not sure if I worked this section correctly but here is my answer:

My answer: -5
∫ (y^2)+ 2y -1 -(-2y + 4)
1

-5
∫ ((y^2) + 4y -5
1

-5
∫ (-y^3)/(3) + 2y^2 -5x
1



-5
∫ (-1/3)(-5^3 -1) + 2 (25 +1) + 5 ( 5 -1)
1

-5
∫ = (341/3)
1