Evaluate the integral below by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using geometry.
�ç6 −6 36−x^2dx
To evaluate the integral ∫ (6 - 6x) dx, we can interpret it in terms of areas by visualizing the region represented by the integral and finding its area using geometry.
First, let's draw a graph to represent the equation y = 6 - 6x by plotting some points:
When x = 0, y = 6 - 6(0) = 6.
When x = 1, y = 6 - 6(1) = 0.
When x = -1, y = 6 - 6(-1) = 12.
Based on these points, we can see that the graph is a downward-sloping straight line that passes through the points (0, 6), (1, 0), and (-1, 12).
Now, let's sketch the region represented by the integral. The region is the area between the x-axis and the graph of y = 6 - 6x. It is a trapezoid with a base of length 2 units and heights of 6 and 12 units.
/¦¯¯¯¯¯¯¯¯¯¯
/ ¦ | |
/ ¦ | |
6 / ¦ | |
/____¦¯¯¯¯¯¯¯¯¯¯
-6 0 6
To find the area of the trapezoid, we can use the formula:
Area = (base1 + base2) * height / 2
In this case, the bases are 6 units and 12 units, and the height is 2 units.
Area = (6 + 12) * 2 / 2
= 18
So, the integral ∫ (6 - 6x) dx represents the area of the region bounded by the x-axis and the graph of y = 6 - 6x, which is equal to 18.