CK-12-Calculus

5.1. Area Between Two Curves http://www.ck12.org Find the area of the region enclosed byx=y^2 andy=x− 6. Solution: Figure 4 As y ...

http://www.ck12.org Chapter 5. Applications of Definite Integrals MEDIA Click image to the left for use the URL below. URL: http ...

5.2. Volumes http://www.ck12.org 5.2 Volumes Learning Objectives Learn the basic concepts of volume and how to compute it with ...

http://www.ck12.org Chapter 5. Applications of Definite Integrals The volume of solid does not necessarily have to be circular. ...

5.2. Volumes http://www.ck12.org If we use the same argument to derive a formula to calculate the area under the curve, let us i ...

http://www.ck12.org Chapter 5. Applications of Definite Integrals Figure 7a Figure 7b Solution: Let they−axis pass through the a ...

5.2. Volumes http://www.ck12.org (^12) b (^12) a= h−y h , b=ah(h−y). Since the cross-sectional area atyisA(y) =b^2 , A(y) =b^2 = ...

http://www.ck12.org Chapter 5. Applications of Definite Integrals Figure 8a Figure 8b Each cross-sectional area can be calculate ...

5.2. Volumes http://www.ck12.org A(x) =π[f(x)]^2. Since the volume is defined as V= ∫b a A(x)dx, the volume of the solid is V= ∫ ...

http://www.ck12.org Chapter 5. Applications of Definite Integrals Figure 9a Figure 9b Example 3: Derive a formula for the volume ...

5.2. Volumes http://www.ck12.org Solution: One way to find the formula is to use the disk method. From your algebra, a circle of ...

http://www.ck12.org Chapter 5. Applications of Definite Integrals Figure 10a Figure 10b Volumes by the Method of Washers(revolut ...

5.2. Volumes http://www.ck12.org Solution: As Figures 11a and 11b show, the volume is Figure 11a ...

http://www.ck12.org Chapter 5. Applications of Definite Integrals Figure 11b From the formula above, V(x) = ∫b a π ([f(x)] (^2) ...

5.2. Volumes http://www.ck12.org What is the volume of the solid generated when the region enclosed byy=√x,y= 3 ,andx=0 is revol ...

http://www.ck12.org Chapter 5. Applications of Definite Integrals Figure 12b Volume By Cylindrical Shells The method of computin ...

5.2. Volumes http://www.ck12.org Figure 13a Figure 13b If we wish to integrate with respect to they−axis, we have to solve forxi ...

http://www.ck12.org Chapter 5. Applications of Definite Integrals Figure 14 Notice however that(r 2 −r 1 )is the thickness of th ...

5.2. Volumes http://www.ck12.org Volume By Cylindrical Shell about thex−Axis Equivalently, if the volume is generated by revolvi ...

http://www.ck12.org Chapter 5. Applications of Definite Integrals Figure 16 Solution: From Figure 15 we can identify the limits ...