# The velocity of an object is given by, f(t) = -t^2 + 4t + 5 a) Find integral_0^4 (-t^2 + 4t +...

## Question:

The velocity of an object is given by, {eq}f(t) = -t^2 + 4t + 5 {/eq}

a) Find {eq}\int_0^4 (-t^2 + 4t + 5)dt {/eq}

b) Find {eq}\int_0^6(-t^2 + 4t + 5)dt {/eq}

c) Find the distance traveled by the object from {eq}t = 0 \ to \ t = 6{/eq}.

## Definite Integral and Distance Traveled :

Here we will use the property of linearity and some basic properties of integration in order to determine the given definite integrals. First of all, we will use the property of linearity and then we will integrate them.

$$\int_{}^{} \; \biggr(a_{1} \; f(x) \pm a_{2} \; g(x) \biggr) \; dx = a_{1} \; \int_{}^{} \; f(x) \; dx \pm a_{2} \; \int_{}^{} g(x) \; dx$$

$$\int_{}^{} \; x^{n} \; dx = \frac {x^{n+1}}{n+1} + C$$

$$\text {Here} \; C \; \text {is the constant of integration.}$$

$$\Rightarrow \text {Distance traveled by an object} \; \; \Rightarrow S(t) = \int_{0}^{t} \; v(t) \; dt$$

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$$\text {The velocity of the particle is given as -}$$

$$v(t) = -t^{2} + 4t + 5$$

Part (a)

I_{1} = \int_{0}^{4} \; \biggr( -t^{2} + 4t + 5...

Evaluating Definite Integrals Using the Fundamental Theorem

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Chapter 16 / Lesson 2
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In calculus, the fundamental theorem is an essential tool that helps explain the relationship between integration and differentiation. Learn about evaluating definite integrals using the fundamental theorem, and work examples to gain understanding.