7.2 Areas in the Plane. How can we find the area between these two curves?

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7.2 Areas in the Plane

How can we find the area between these two curves?

Consider a very thin vertical strip. The length of the strip is: or Since the width of the strip is a very small change in x, we could call it dx.

Since the strip is a long thin rectangle, the area of the strip is: If we add all the strips (Riemann sums), we get: as the width of the strips approach 0.

The formula for the area between curves using vertical strips (dx) is: Be sure to subtract the top function – the bottom function.

If we try vertical strips, we have to integrate in two parts: We can find the same area using a horizontal strip. Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y.

We can find the same area using a horizontal strip. Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y. length of strip width of strip

The formula for the area between curves using horizontal strips (dy) is: Be sure to subtract function to the right – function to the left

General Strategy for Area Between Curves: 1 Decide on vertical or horizontal strips. (Pick whichever is easier to write formulas for the length of the strip, and/or whichever will let you integrate fewer times.) Sketch the curves. 2 3 Write an expression for the area of the strip. (If the width is dx, the length must be in terms of x. If the width is dy, the length must be in terms of y. 4 Find the limits of integration. (If using dx, the limits are x values; if using dy, the limits are y values.) 5 Integrate to find area.