Exercises 8.1In Exercises 1 through 20, find the area of the region bounded by the given curves. In each problem do the following: (a)Draw a figure showing the region and a rectangular element of area; (b) express the area of the region as the limit of aRiemann sum; (e) find the limit in part (b) by evaluating a definite integral by the fundamental theorem of the calculus.í. x* --y; y--4L yt =-x; x=-2; x =-4330APPLICATIONS OF THE DEFINITE INTEGRAL3. x* + y + 4 = 0; y =-8. Take the elements of area perpendicular to the y axis.4. The same region as in Exercise 3. Take the elements of area parallel to the y axis.5. x= 2y*; x = 0, y=-2d y = 4x; x = 0; y =-29. yi = x– 1; x= 312. x= 4- y"; x =4 – 4y15. x= y" – 2; x = 6 – y*13. y =2-r; y =-I10. y = x*; x* = 18 - y13. y = x*; x– 3y + 4= 08. y = x*; y = xk1. y = Vx; y = x14. xy = y? – 1; x= 1; y = 1; y= 416. x= y* - y; x = y – y18. 3y = x" – 2x* – 15x; y = x" – 4xª - 11x + 3017. y = 2x" – 3x* –- 9x; y = x³ – 2x – 3x19. y = |x|, y = x² – 1, x =-1, x=120. y = |x + 1| + |x\, y = 0, x = -2, x =321. Find by integration the area of the triangle having vertices at (5, 1), (1, 3), and (-1,–2).22. Find by integration the area of the triangle having vertices at (3, 4), (2, 0), and (0, 1).23. Find the area of the region bounded by the curve x-x* + 2xy – y² = 0 and the line x= 4. (HINT: Solve the cubic equa-tion for y in terms of x, and express y as two functions of x.)24. Find the area of the region bounded by the three curves y = x*, x = y', and x + y = 2.26. Find the area of the region bounded by the three curves y = x*, y = 8 – xª, and 4x – y + 12 =0.26. Find the area of the region above the parabola x* = 4py and inside the triangle formed by the x axis and the lines y = x+ 8p and y =-x + 8p.

The objective is to find the area of region between the curves a) Sketch the curve. b) Express…

Exercises 8.1 In Exercises 1 through 20, find the area of the region bounded by the given curves. In each problem do the following: (a) Draw a figure showing the region and a rectangular element of area; (b) express the area of the region as the limit of a Riemann sum; (c) find the limit in part (b) by evaluating a definite integral by the fundamental theorem of the calculus. í. --y; y--4 y--x; x--2; =-4

Exercises 8.1 In Exercises 1 through 20, find the area of the region bounded by the given curves. In each problem do the following: (a) Draw a figure showing the region and a rectangular element of area; (b) express the area of the region as the limit of a Riemann sum; (c) find the limit in part (b) by evaluating a definite integral by the fundamental theorem of the calculus. í. --y; y--4 y--x; x--2; =-4

Advanced Engineering Mathematics

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Chapter2: Second-order Linear Odes

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

Exercises 8.1
In Exercises 1 through 20, find the area of the region bounded by the given curves. In each problem do the following: (a)
Draw a figure showing the region and a rectangular element of area; (b) express the area of the region as the limit of a
Riemann sum; (e) find the limit in part (b) by evaluating a definite integral by the fundamental theorem of the calculus.
í. x* --y; y--4
L yt =-x; x=-2; x =-4
330
APPLICATIONS OF THE DEFINITE INTEGRAL
3. x* + y + 4 = 0; y =-8. Take the elements of area perpendicular to the y axis.
4. The same region as in Exercise 3. Take the elements of area parallel to the y axis.
5. x= 2y*; x = 0, y=-2
d y = 4x; x = 0; y =-2
9. yi = x– 1; x= 3
12. x= 4- y"; x =4 – 4y
15. x= y" – 2; x = 6 – y*
13. y =2-r; y =-I
10. y = x*; x* = 18 - y
13. y = x*; x– 3y + 4= 0
8. y = x*; y = x
k1. y = Vx; y = x
14. xy = y? – 1; x= 1; y = 1; y= 4
16. x= y* - y; x = y – y
18. 3y = x" – 2x* – 15x; y = x" – 4xª - 11x + 30
17. y = 2x" – 3x* –- 9x; y = x³ – 2x – 3x
19. y = |x|, y = x² – 1, x =-1, x=1
20. y = |x + 1| + |x\, y = 0, x = -2, x =3
21. Find by integration the area of the triangle having vertices at (5, 1), (1, 3), and (-1,–2).
22. Find by integration the area of the triangle having vertices at (3, 4), (2, 0), and (0, 1).
23. Find the area of the region bounded by the curve x-x* + 2xy – y² = 0 and the line x= 4. (HINT: Solve the cubic equa-
tion for y in terms of x, and express y as two functions of x.)
24. Find the area of the region bounded by the three curves y = x*, x = y', and x + y = 2.
26. Find the area of the region bounded by the three curves y = x*, y = 8 – xª, and 4x – y + 12 =0.
26. Find the area of the region above the parabola x* = 4py and inside the triangle formed by the x axis and the lines y = x
+ 8p and y =-x + 8p.

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