Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫¹₁/₂ (t⁻³ ― 8) dt

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Step 1: Identify the integral to be evaluated: ∫_1/2¹ (t⁻³ − 8) dt. This is a definite integral, meaning we will evaluate it over the interval [1/2, 1].

Step 2: Break the integral into two parts for clarity: ∫_1/2¹ t⁻³ dt − ∫_1/2¹ 8 dt. This uses the property of linearity of integrals.

Step 3: Use the Fundamental Theorem of Calculus to find the antiderivative of each term. For t⁻³, the antiderivative is

-t

^-2/2. For the constant 8, the antiderivative is 8t.

Step 4: Apply the limits of integration [1/2, 1] to each antiderivative. For the first term, evaluate

-t

^-2/2 at t = 1 and t = 1/2, and subtract the results. For the second term, evaluate 8t at t = 1 and t = 1/2, and subtract the results.

Step 5: Combine the results from both parts to get the final value of the definite integral. Simplify the expressions as needed.

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Definite Integrals

Definite integrals represent the signed area under a curve defined by a function over a specific interval. They are calculated using the limits of integration, which specify the range over which the area is computed. The result of a definite integral is a numerical value that reflects this area, and it is denoted as ∫[a, b] f(t) dt, where [a, b] are the limits of integration.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with integration, providing a method to evaluate definite integrals. It states that if F is an antiderivative of f on an interval [a, b], then ∫[a, b] f(t) dt = F(b) - F(a). This theorem allows us to compute the value of a definite integral by finding the antiderivative of the integrand and evaluating it at the boundaries.

Antiderivatives

An antiderivative of a function f(t) is another function F(t) such that F'(t) = f(t). Finding antiderivatives is essential for applying the Fundamental Theorem of Calculus, as it allows us to determine the area under the curve represented by f(t). Common techniques for finding antiderivatives include power rule, substitution, and integration by parts, depending on the complexity of the function.

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus∫¹₁/₂ (t⁻³ ― 8) dt