SYMMETRY Even and Odd Suppose f is continuous on [-a, a] and even Suppose f is continuous on [-a, a] and odd

Even and Odd Term-102

Even and Odd Term-091

Even and Odd Term-103

SYMMETRY Even and Odd Suppose f is continuous on [-a, a] and even Suppose f is continuous on [-a, a] and odd Example

Area between two curves AREAS BETWEEN CURVES Area between two curves

Example: Note: AREAS BETWEEN CURVES Area between two curves Find the area of the region bounded by the curves and Note: Both right-hand and left-hand boundary are lines

Area between two curves AREAS BETWEEN CURVES Area between two curves Note: Both right-hand and left-hand boundary reduce to a point

1) Find the intersection points AREAS BETWEEN CURVES Note: The value of a and b (not given) Steps: 1) Find the intersection points 2) Write the area as an integral

Area between two curves AREAS BETWEEN CURVES Area between two curves

AREAS BETWEEN CURVES T-092

AREAS BETWEEN CURVES T-102

AREAS BETWEEN CURVES

Area between two curves AREAS BETWEEN CURVES Area between two curves Note: Note: right-hand boundary = ?? left-hand boundary = ?? right-hand boundary = ?? left-hand boundary = ?? Note: Note: Top boundary = ?? Bottom boundary = ?? Top boundary = ?? Bottom boundary = ??

Area between two curves AREAS BETWEEN CURVES Area between two curves Some regions are best treated by regarding x as a function of y.

AREAS BETWEEN CURVES Some regions are best treated by regarding x as a function of y.

1) Rewrite the function as x in terms of y AREAS BETWEEN CURVES Steps: 1) Rewrite the function as x in terms of y 2) Find the intersection points

AREAS BETWEEN CURVES We could have found the area in by integrating with respect to x instead of y, but the calculation is much more involved.

AREAS BETWEEN CURVES T-092

AREAS BETWEEN CURVES