Question Statement

Find:

${\int }_3^{-2}f(x)dx$

given that:

${\int }_{-2}^{1}f(x)dx=1\text{and}{\int }_1^{3}f(x)dx=-5$

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Background and Explanation

This problem tests your understanding of definite integral properties, specifically how to handle reversed limits of integration and how to combine integrals over adjacent intervals. You will need to apply the interval addition property to break the integral into pieces where values are known.

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Solution

We are asked to find ${\int }_3^{-2}f(x)dx$, but notice that the upper limit ($-2$) is less than the lower limit ($3$). This is an integral with reversed bounds.

Step 1: Reverse the limits of integration

When the upper limit is smaller than the lower limit, we can reverse the bounds by introducing a negative sign:

${\int }_3^{-2}f(x)dx=-{\int }_{-2}^{3}f(x)dx$

This property allows us to work with a standard integral from $-2$ to $3$.

Step 2: Split the integral using the interval addition property

We can break the integral ${\int }_{-2}^{3}f(x)dx$ into two parts at any point between $-2$ and $3$. Since we know values at $x=1$, we split there:

$-{\int }_{-2}^{3}f(x)dx=-\left[{\int }_{-2}^{1}f(x)dx+{\int }_1^{3}f(x)dx\right]$

Step 3: Substitute the given values

We are given that ${\int }_{-2}^{1}f(x)dx=1$ and ${\int }_1^{3}f(x)dx=-5$. Substituting these values:

$=-[1+(-5)]$

Step 4: Simplify the arithmetic

First, simplify inside the brackets: $=-[1-5]$

$=-[-4]$

Finally, distribute the negative sign: $=4$

Therefore:

${\int }_3^{-2}f(x)dx=4$

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Key Formulas or Methods Used

• Reversal of Limits: ${\int }_a^{b}f(x)dx=-{\int }_b^{a}f(x)dx$
• Interval Addition Property: ${\int }_a^{c}f(x)dx={\int }_a^{b}f(x)dx+{\int }_b^{c}f(x)dx$ (where $a<b<c$)
• Definite Integral Evaluation: Substituting known values and performing arithmetic operations

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Summary of Steps

1. Apply the reversal of limits property to convert ${\int }_3^{-2}f(x)dx$ to $-{\int }_{-2}^{3}f(x)dx$
2. Use the interval addition property to split ${\int }_{-2}^{3}f(x)dx$ into ${\int }_{-2}^{1}f(x)dx+{\int }_1^{3}f(x)dx$
3. Substitute the given values $1$ and $-5$ into the expression
4. Simplify: $-[1+(-5)]=-[-4]=4$