6FExample2

Finding Area Between curve and y-axis

Example 1

a) Find the area bounded by the curve **//f(x) = log e x//** and the **y-axis** and between the lines **//y = log e 4//** and **y = 0**.

math \text{Function with x as the subject } \; \Rightarrow \; x=e^y math

math \text{Area } = \displaystyle{ \int \limits_{y=0}^{y=log_e4} e^y \; dy} math . math . \qquad = \Big[ e^y \Big]_{y=0}^{y=log_e4} math . math . \qquad =e^{log_e4} - e^0 math . math \\ . \qquad = 4 - 1 \\. \\ . \qquad = 3 \; \text{ square units} math

b) Hence find the area bounded by the curve **//f(x) = log e x//**, the **x-axis** and the line **x = 4**

From the graph we see that this is math \text{Area } = \displaystyle{ \int\limits_{x=1}^{x=4} log_ex \; dx} math

Inspection of the graph shows that the required area is the remainder of the rectangle created by the axes and the lines **//y = log e 4//** and **x = 4**

From (a) we know that the reverse area equals 3.

math \text{Area(rectangle) } = 4log_e4 math

Thus math \text{Area } = 4log_e4 - 3 \; \text{ square units} math

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