Solved Worksheet 1: Review of Definite integrals Keywords: | Chegg.com - Free Printable

We are given two definite integrals:

$$
\int_1^4 f(x)\,dx = 8 \quad \text{and} \quad \int_1^6 f(x)\,dx = 5
$$

We are to evaluate the following four integrals using properties of definite integrals.

---

(a) $\displaystyle \int_4^1 -3f(x)\,dx$

We use two properties:

1. Reversing limits:
$$
\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx
$$
So,
$$
\int_4^1 -3f(x)\,dx = -\int_1^4 -3f(x)\,dx
$$

2. Constant multiple rule:
$$
\int_a^b c f(x)\,dx = c \int_a^b f(x)\,dx
$$

So:
$$
\int_4^1 -3f(x)\,dx = -\left(-3 \int_1^4 f(x)\,dx\right) = 3 \int_1^4 f(x)\,dx
$$

Now substitute the known value:
$$
= 3 \cdot 8 = 24
$$

✔ Answer (a): $ \boxed{24} $

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(b) $\displaystyle \int_4^6 5f(x)\,dx$

We need to relate this to the given integrals. We know:

- $ \int_1^4 f(x)\,dx = 8 $
- $ \int_1^6 f(x)\,dx = 5 $

Using the additive property of integrals:
$$
\int_1^6 f(x)\,dx = \int_1^4 f(x)\,dx + \int_4^6 f(x)\,dx
$$

Substitute known values:
$$
5 = 8 + \int_4^6 f(x)\,dx
\Rightarrow \int_4^6 f(x)\,dx = 5 - 8 = -3
$$

Now compute:
$$
\int_4^6 5f(x)\,dx = 5 \int_4^6 f(x)\,dx = 5 \cdot (-3) = -15
$$

✔ Answer (b): $ \boxed{-15} $

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(c) $\displaystyle \int_4^6 f(x)\,dx$

We already computed this in part (b):

$$
\int_4^6 f(x)\,dx = -3
$$

✔ Answer (c): $ \boxed{-3} $

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(d) $\displaystyle \int_6^4 2f(x)\,dx$

Again, reverse the limits:
$$
\int_6^4 2f(x)\,dx = -\int_4^6 2f(x)\,dx
$$

From part (b), we know:
$$
\int_4^6 2f(x)\,dx = 2 \int_4^6 f(x)\,dx = 2 \cdot (-3) = -6
$$

So:
$$
\int_6^4 2f(x)\,dx = -(-6) = 6
$$

✔ Answer (d): $ \boxed{6} $

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✔ Final Answers:

(a) $ \boxed{24} $
(b) $ \boxed{-15} $
(c) $ \boxed{-3} $
(d) $ \boxed{6} $