Question

In the following problems find the total area bounded by the graph of the function given and the x-axis in the given interval. 1.y=x-1;[-1,1]

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Darrell

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64 Answers

Step 1: Identify the function and interval: y = x - 1; \quad [-1, 1]

Step 2: Integrate the function over the interval.

\int_{-1}^{1} (x - 1) \, dx

Step 3: Find the antiderivative:

\int (x - 1) \, dx = \frac{x^2}{2} - x

Step 4: Evaluate from -1 to 1:

\left[ \frac{x^2}{2} - x \right]_{-1}^{1} = \left( \frac{1^2}{2} - 1 \right) - \left( \frac{(-1)^2}{2} - (-1) \right)

Step 5: Simplify the terms inside the brackets:

\left( \frac{1}{2} - 1 \right) - \left( \frac{1}{2} + 1 \right) = -\frac{1}{2} - \frac{3}{2} = -2

Step 6: Take the absolute value of the result to get the total area (since area cannot be negative):

\left| -2 \right| = 2

Therefore, the answer is:

2

Step 2: Integrate the function over the interval.

Step 3: Find the antiderivative:

Step 4: Evaluate from -1 to 1:

Step 5: Simplify the terms inside the brackets:

Step 6: Take the absolute value of the result to get the total area (since area cannot be negative):

Therefore, the answer is:

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