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Chapter 5: Problem 74

Use geometry to evaluate the following integrals. $$\int_{1}^{6}(3 x-6) d x$$

Short Answer

Expert verified
Answer: The value of the integral is 9.

Step by step solution

01

Identify the geometrical shapes

To find the area under the curve, first, let's graph the function \(f(x) = 3x - 6\). Looking at the graph, we can clearly see that between \(x=1\) and \(x=6\), the area under the curve can be split into a rectangle and a triangle.
02

Identify the vertices of the geometrical shapes

Now, let's find the vertices of the rectangle and the triangle. We need to find the points where \(f(x) = 0\) in the given interval. $$f(x) = 3x - 6 = 0$$ $$x = 2$$ So the vertices of the rectangle are \((1, 0)\), \((2, 0)\), \((1, -3)\), and \((2, -3)\). And the vertices of the triangle are \((2, 0)\), \((6, 0)\), and \((2, -3)\).
03

Calculate the area of the rectangle

The area of a rectangle is given by the formula: Area \(= base \cdot height\). In our case, the base is 1 unit (\((2 - 1)\)) and the height is 3 units (the absolute value of \(f(1)\)). So, the area of the rectangle is: $$A_{rectangle} = 1 \cdot 3 = 3$$
04

Calculate the area of the triangle

The area of a triangle is given by the formula: Area \(= 0.5 \cdot base \cdot height\). In our case, the base is 4 units (\((6 - 2)\)) and the height is 3 units (the absolute value of \(f(2)\)). So, the area of the triangle is: $$A_{triangle} = 0.5 \cdot 4 \cdot 3 = 6$$
05

Add the areas and find the value of the integral

Now that we have calculated the areas of both the geometrical shapes, we can add them together to find the value of the integral. $$\int_{1}^{6}(3 x-6) d x = A_{rectangle} + A_{triangle} = 3 + 6 = 9$$

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