Even and Odd SYMMETRY Suppose f is continuous

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

Even and Odd SYMMETRY 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-102

Even and Odd Term-091

Even and Odd Term-091

Even and Odd Term-103

Even and Odd Term-103

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

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

AREAS BETWEEN CURVES Area between two curves

AREAS BETWEEN CURVES Area between two curves

AREAS BETWEEN CURVES Area between two curves Example: Find the area of the region

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

AREAS BETWEEN CURVES Area between two curves Note: Both right-hand lefthand boundary reduce to

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

AREAS BETWEEN CURVES Note: The value of a and b (not given) Steps: 1)

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

AREAS BETWEEN CURVES Area between two curves

AREAS BETWEEN CURVES Area between two curves

AREAS BETWEEN CURVES T-092

AREAS BETWEEN CURVES T-092

AREAS BETWEEN CURVES T-102

AREAS BETWEEN CURVES T-102

AREAS BETWEEN CURVES

AREAS BETWEEN CURVES

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

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

AREAS BETWEEN CURVES Area between two curves Some regions are best treated by regarding

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

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

AREAS BETWEEN CURVES Steps: 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

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 T-092

AREAS BETWEEN CURVES

AREAS BETWEEN CURVES