MAT133Y5 Chapter Notes - Chapter 8: Constant Function
CHAPTER 8- APPLICATIONS OF INTEGRATION
SUMMARY
Area Computations
One of the primary motivations for developing the theory of integral
calculus is to actually compute areas. The area of some objects are easy
to compute; such as rectangles, triangles, parallelograms, and even
trapezoids. Our ability to find formulas for the area of these shapes
hinges upon the fact that they are constructed with straight lines, and so
may be related to rectangles.
Rectangles:
Consider a rectangle with height h and width w. Let f : [0,w] → R be the
constant function f(x) = h. The area under the graph of f is precisely the
area of the rectangle,
Triangles:
Consider a triangle with base b and height h. Define the function f : [0,h]
→ R by f(x) = bx, which is a straight
h
line with height f(h) = b. The area under f is the area of the desired
rectangle,
Circles:
Let r > 0 be the radius of our circle. We know that the formula of a circle
is given by x2 +y2 = r2. We cannot write the circle as a function though,
but by writing y = √r2 − x2 and integrating on [0, r] we can determine a
quarter of the area of the circle. If we multiply by 4 at the end we will
get the full area of the circle.
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Document Summary
One of the primary motivations for developing the theory of integral calculus is to actually compute areas. The area of some objects are easy to compute; such as rectangles, triangles, parallelograms, and even trapezoids. Our ability to find formulas for the area of these shapes hinges upon the fact that they are constructed with straight lines, and so may be related to rectangles. Consider a rectangle with height h and width w. let f : [0,w] r be the constant function f(x) = h. the area under the graph of f is precisely the area of the rectangle, Consider a triangle with base b and height h. define the function f : [0,h] R by f(x) = bx, which is a straight h line with height f(h) = b. The area under f is the area of the desired rectangle, Let r > 0 be the radius of our circle.
