Area & Volume Chapter 6.1 & 6.2 February 20, 2007

Back to Area: We can calculate the area between the x-axis and a continuous function f on the interval [a,b] using the definite integral: Where f(xi*) is the height of a rectangle and ∆x is the width of that rectangle. {(b-a)/n (n is the number of rectangles)} Remember that the area above the axis is positive and the area below is negative.

Set up the integral needed to find the area of the region bounded by: and the x-axis.

Set up the integral needed to find the area of the region bounded by: , the x-axis on [0,2].

Area bounded by two curves Suppose you have 2 curves, y = f(x) and y = g(x) Area under f is: Area under g is:

Superimposing the graphs, we look at the area bounded by the two functions: (top - bottom)*∆x

The area bounded by two functions can be found:

Find the area of the region between the two functions: and Bounds? [-1,2] Top Function? Bottom Function? Area? = 9

Find the area bounded by the curves: and Solve for bounds:

Find the area bounded by the curves: and Sketch the graph: (top - bottom)*∆x

Find the area of the region determined by the curves: and Bounds? In terms of y: [-2,4] Points (-1,-2) & (5,4) Graph? Solve for y:

Find the area of the region determined by the curves: and Need 2 Integrals! One from -3 to -1 and the other from -1 to 5. Area?

Horizontal Cut instead: Bounds? In terms of y: [-2,4] Right Function? Left Function? Area? = 18

In General: Vertical Cut: Horizontal Cut:

Find the Area of the Region bounded by and Bounds? [0,1] Top Function? Bottom Function? Area?

Find the Area of the Region bounded by , , and the y-axis Bounds? [0,π/4] and [π/4, π/2] [0,π/4] Top Function? Bottom Function? Area?

Find the Area of the Region bounded by , , and the y-axis Bounds? [0,π/4] and [π/4, π/2] Top Function? Bottom Function? Area?

Find the area of the Region bounded by Bounds? Interval is from -2, 5 Functions intersect at x = -1 and x = 3 Graph? Top function switches 3 times! This calculation requires 3 integrals!

Find the area of the Region bounded by

Find T so the area between y = x2 and y = T is 1/2. Bounds? Top Function? Bottom Function? Area? Taking advantage of Symmetry Area must equal 1/2: Ans:

In-class Integrate: Set up an integral to find the area of the region bounded by:

In-class Integrate: Set up an integral to find the area of the region bounded by:

Volume & Definite Integrals We used definite integrals to find areas by slicing the region and adding up the areas of the slices. We will use definite integrals to compute volume in a similar way, by slicing the solid and adding up the volumes of the slices. For Example………………

Find the volume of the solid whose bottom face is the circle and every cross section perpendicular to the x-axis is a square. [-1,1] Bounds? Top Function? Bottom Function? Length?

Find the volume of the solid whose bottom face is the circle and every cross section perpendicular to the x-axis is a square. We use this length to find the area of the square. Length? Area? Volume?

Find the volume of the solid whose bottom face is the circle and every cross section perpendicular to the x-axis is a square. What does this shape look like? Volume?

Volumes: We will be given a “boundary” for the base of the shape which will be used to find a length. We will use that length to find the area of a figure generated from the slice . The dy or dx will be used to represent the thickness. The volumes from the slices will be added together to get the total volume of the figure.

Find the volume of the solid whose bottom face is the circle and every cross section perpendicular to the x-axis is a circle with diameter in the plane. Length? Area? Radius: Volume?

Using the half circle [0,1] as the base slices perpendicular to the x-axis are isosceles right triangles. Bounds? [0,1] Length? Area? Volume? Visual?