Evaluate the integral by interpreting it in terms of areas. 9 0 1 3 x − 2 dx

General
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evaluate the integral by interpreting it in terms of areas. 9 0 1 3 x − 2 dx

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To evaluate the integral ∫(9 + 0 + 1 + 3x - 2)dx by interpreting it in terms of areas, we can break it down into simpler geometric shapes and calculate their areas.

The given integral can be rewritten as ∫(3x + 8)dx. Let’s evaluate it step by step:

  1. Consider the first term 3x. This represents a line with a slope of 3. Integrating a line corresponds to finding the area under the line. In this case, the area under the line 3x can be calculated as 1/2 * base * height, where the base is the difference in x-values and the height is the y-value (3x).

  2. Integrating the line 3x gives us the area of a triangle. The base of the triangle is from x = 0 to x = 1 (1 - 0 = 1), and the height is 3x. So the area of the triangle is 1/2 * 1 * 3 = 1.5.

  3. Now, let’s consider the second term, 8. This is a constant, which represents a horizontal line on the y-axis. The integral of a constant represents the area of a rectangle.

  4. The rectangle has a width equal to the base of the triangle (1) and a height of 8. So the area of the rectangle is 1 * 8 = 8.

  5. Adding the areas of the triangle and the rectangle, we get the total area under the curve given by the integral. The total area is 1.5 + 8 = 9.5.

Therefore, the value of the integral of (9 + 0 + 1 + 3x - 2)dx, interpreted in terms of areas, is 9.5.