![Region S Between Two Curves
▶ Let y = f (x) and y = g(x)
be two curves such that
f (x) ≥ g(x) between the
vertical lines x = a and
x = b, and f and g are
continuous on [a, b].
▶ Let S be the region
enclosed by these two
curves and A the area of S.](https://image.slidesharecdn.com/calcii61summer2022-220704135623-11350494/85/Section-6-1-pdf-4-320.jpg)
![Computing the Area of S - Approximating with Rectangles
▶ Divide S into n strips of
equal width and
approximate the ith strip by
a rectangle with base △x
and height f (x∗
i ) − g(x∗
i ).
▶ Here x∗
i is any value in the
interval [xi−1, xi ].](https://image.slidesharecdn.com/calcii61summer2022-220704135623-11350494/85/Section-6-1-pdf-5-320.jpg)
![Computing the Area of R - Approximating with Rectangles
▶ The n rectangles have
equal width △x = b−a
n
height = f (x∗
i ) − g(x∗
i ).
▶ The area of a rectangle is
[height]·[width]=
[f (x∗
i ) − g(x∗
i )]△x
A ≈
n
X
i=1
[f (x∗
i ) − g(x∗
i )]△x for large values of n =⇒
A = lim
n→∞
n
X
i=1
[f (x∗
i ) − g(x∗
i )]△x =
Z b
a
[f (x) − g(x)]dx](https://image.slidesharecdn.com/calcii61summer2022-220704135623-11350494/85/Section-6-1-pdf-7-320.jpg)
![Area Between Curves
▶ So if f (x) ≥ g(x) for all a ≤ x ≤ b, then the area of the
region between f and g on the interval a ≤ x ≤ b is
A =
Z b
a
[f (x) − g(x)]dx
▶ Denoting the top function by yTop or yT , and the bottom
function by yBottom or yB we have
A =
Z b
a
[yT − yB]dx](https://image.slidesharecdn.com/calcii61summer2022-220704135623-11350494/85/Section-6-1-pdf-8-320.jpg)
![Suppose we are asked to find the area between two curves f and g
where f (x) ≥ g(x) for some values of x and g(x) ≥ f (x) for other
values of x.
Let Ai be the area of the region
Si for i = 1, 2, 3.
S = S1 ∪ S2 ∪ S3
A = A1 + A2 + A3
=
Z x1
a
[f (x) − g(x)]dx + . . .
Z x2
x1
[g(x) − f (x)]dx + . . .
Z b
x2
[f (x) − g(x)]dx](https://image.slidesharecdn.com/calcii61summer2022-220704135623-11350494/85/Section-6-1-pdf-13-320.jpg)
![Some regions are best treated by regarding x as a function of y.
If we have a region bounded by
the following,
x = f (y)
x = g(y)
y = c
y = d
where f and g are continuous
and f (y) ≥ g(y) for c ≤ y ≤ d
then its area is
A =
Z d
c
[f (y) − g(y)]dy](https://image.slidesharecdn.com/calcii61summer2022-220704135623-11350494/85/Section-6-1-pdf-18-320.jpg)
Section 6.1.pdf
- 1. Chapter 6: Applications of Integration Section 6.1: Areas Between Curves Alea Wittig SUNY Albany
- 2. Outline Definition of Area Between Curves Examples Region S with Alternating Top and Bottom Curves Example Region S with Left and Right Curves Examples
- 3. Definition of Area Between Curves
- 4. Region S Between Two Curves ▶ Let y = f (x) and y = g(x) be two curves such that f (x) ≥ g(x) between the vertical lines x = a and x = b, and f and g are continuous on [a, b]. ▶ Let S be the region enclosed by these two curves and A the area of S.
- 5. Computing the Area of S - Approximating with Rectangles ▶ Divide S into n strips of equal width and approximate the ith strip by a rectangle with base △x and height f (x∗ i ) − g(x∗ i ). ▶ Here x∗ i is any value in the interval [xi−1, xi ].
- 6. Computing the Area of S - Approximating with Rectangles → ▶ The sum of the areas of the n rectangles will approximate the exact area A of the region S for large n. ▶ As we take n → ∞, the sum of the areas of the rectangles will give the exact area A of S.
- 7. Computing the Area of R - Approximating with Rectangles ▶ The n rectangles have equal width △x = b−a n height = f (x∗ i ) − g(x∗ i ). ▶ The area of a rectangle is [height]·[width]= [f (x∗ i ) − g(x∗ i )]△x A ≈ n X i=1 [f (x∗ i ) − g(x∗ i )]△x for large values of n =⇒ A = lim n→∞ n X i=1 [f (x∗ i ) − g(x∗ i )]△x = Z b a [f (x) − g(x)]dx
- 8. Area Between Curves ▶ So if f (x) ≥ g(x) for all a ≤ x ≤ b, then the area of the region between f and g on the interval a ≤ x ≤ b is A = Z b a [f (x) − g(x)]dx ▶ Denoting the top function by yTop or yT , and the bottom function by yBottom or yB we have A = Z b a [yT − yB]dx
- 9. Examples
- 10. Example 1 Find the area of the region bounded above by y = ex , below by y = x and on the sides by x = 0 and x = 1 Video Solution 1
- 11. Example 2 Find the area of the region enclosed by the parabolas y = x2 and y = 2x − x2. Video Solution 2
- 12. Region S with Alternating Top and Bottom Curves
- 13. Suppose we are asked to find the area between two curves f and g where f (x) ≥ g(x) for some values of x and g(x) ≥ f (x) for other values of x. Let Ai be the area of the region Si for i = 1, 2, 3. S = S1 ∪ S2 ∪ S3 A = A1 + A2 + A3 = Z x1 a [f (x) − g(x)]dx + . . . Z x2 x1 [g(x) − f (x)]dx + . . . Z b x2 [f (x) − g(x)]dx
- 14. In general, since |f (x) − g(x)| = ( f (x) − g(x) if f (x) ≥ g(x) g(x) − f (x) if g(x) ≥ f (x) we have that the area between the curves y = f (x) and y = g(x) and between x = a and x = b is A = Z b a |f (x) − g(x)|dx
- 15. Example
- 16. Example 3 Find the area bounded by the curves y = sin x and y = cos x, x = 0, and x = π/2. Video Solution 3
- 17. Region S with Left and Right Curves
- 18. Some regions are best treated by regarding x as a function of y. If we have a region bounded by the following, x = f (y) x = g(y) y = c y = d where f and g are continuous and f (y) ≥ g(y) for c ≤ y ≤ d then its area is A = Z d c [f (y) − g(y)]dy
- 19. Examples
- 20. Example 4 Find the area enclosed by the curves x = 9 − y2 and x = y2 − 9 Video Solution 4
- 21. Example 5 Find the area of the region (in quadrant 1) enclosed by the curves y = 1/x, y = x, and y = 1 4x using a. x as the variable of integration and b. y as the variable of integration. Video Solution 5