The Fundamental Theorem of Calculus and Definite Integrals

The Fundamental Theorem of Calculus and Definite Integrals: AP Calculus AB/BC Study Guide

Introduction

Hello, calculus enthusiasts! Prepare to embark on a thrilling mathematical adventure with the Fundamental Theorem of Calculus (FTOC). This key topic is like the Swiss Army knife of calculus, seamlessly connecting differentiation and integration. Both parts of the theorem reveal a beautiful relationship between a definite integral and its antiderivative. So, grab your calculators and let’s dive in! 🧮✨

Fundamental Theorem of Calculus - Part 1

The first part of the FTOC is like the ultimate unboxing video but for calculus. It states that if you take the derivative of an integral, you get back the integrand (the inside function). Let's write this out:

g(x) = ∫ from a to x of f(t) dt

Differentiating both sides, we get:

g′(x) = f(x)

What the FTOC is telling us here is basically like saying, "Hey, differentiation and integration are inverse operations. If you differentiate an integral, you’ll get back your original function, f(x)." It’s the mathematical equivalent of undoing a knot! 🎁✂️

Example: Normal FTOC

Find ( g'(x) ) if ( g(x) = \int_{2}^{x} 5t^4 dt ).

Using the FTOC, we find:

g'(x) = 5x^4

Example: Upper Bound Change

Find ( g'(x) ) if ( g(x) = \int_{3}^{2x} 5t^4 dt ).

Here, the upper bound is not just ( x ), but ( 2x ). This means we apply the chain rule:

g'(x) = 5(2x)^4 * (d/dx)(2x) = 80x^4

Visual Learners Unite!

Imagine the function f(t) as a river flowing steadily. Taking the integral is like collecting water in a bucket from point a to point x. The FTOC Part 1 tells us that if we measure the rate at which water flows out of the bucket (by differentiating), we get back the river's flow rate (f(x)).

Fundamental Theorem of Calculus - Part 2

The second part of the FTOC is like having a cheat code for calculating the area under a curve. It allows you to find the definite integral of a function over [a, b] without actually doing the integration (woo-hoo!). If a function f(x) is continuous on the interval [a, b] and F(x) is an antiderivative of f(x), then:

∫ from a to b of f(x) dx = F(b) - F(a)

Imagine you've got a magical blender. You blend the function from a to b, and then pour out two glasses, one called F(b) and the other F(a). Subtracting the content of glass F(a) from glass F(b) gives you the integral. 🍹🍹➖

Steps to Solving a Definite Integral with FTOC 2

1. Find the antiderivative F(x) of f(x).
2. Plug the upper bound b into F(x).
3. Plug the lower bound a into F(x).
4. Subtract F(a) from F(b), and viola!

Example: Solving with FTOC Part 2

Evaluate ( \int_{0}^{5} e^x dx ).

1. Find the antiderivative of ( e^x ), which is ( e^x ).
2. Evaluate at the upper bound: ( e^5 ).
3. Evaluate at the lower bound: ( e^0 = 1 ).
4. Subtract: ( e^5 - 1 ).

Final answer: ( e^5 - 1 )

FTOC Part 2 Practice Problems

1. Calculate the integral of ( f(x) = 3x^2 ) from ( a = 1 ) to ( b = 4 ).
2. Evaluate the integral ( \int_{0}^{2} (5x - 2) dx ).
3. Find the definite integral of ( f(x) = 2e^x ) from ( a = 0 ) to ( b = 1 ).
4. Evaluate ( \int_{2}^{3} \frac{1}{x} dx ).

FTOC Part 2 Solutions

Question 1 Solution

( \int_{1}^{4} 3x^2 dx )

Antiderivative: ( F(x) = x^3 )

Evaluate: ( F(4) = 4^3 = 64 ) and ( F(1) = 1^3 = 1 )

Subtract: ( 64 - 1 = 63 ) 🎉

Question 2 Solution

( \int_{0}^{2} (5x - 2) dx )

Antiderivative: ( \frac{5}{2} x^2 - 2x )

Evaluate: ( \left( \frac{5}{2}(2)^2 - 2(2) \right) ) and ( 0 )

Subtract: ( \left(10 - 4\right) - 0 = 6 )

Question 3 Solution

( \int_{0}^{1} 2e^x dx )

Antiderivative: ( 2e^x )

Evaluate: ( 2e^1 = 2e ) and ( 2e^0 = 2 )

Subtract: ( 2e - 2 )

Question 4 Solution

( \int_{2}^{3} 1/x dx )

Antiderivative: ( \ln|x| )

Evaluate: ( \ln 3 ) and ( \ln 2 )

Subtract: ( \ln 3 - \ln 2 )

Closing Remarks

With practice, you will master the art of evaluating any definite integrals thrown your way! The FTOC is not just a tool; it’s a cardinal rule in the fascinating world of calculus. So, keep practicing, and may your integrals always be definite and your calculations precise! 🌟🧠

Go forth and conquer those integrals like a calculus superhero!

All Study GuidesAP Calculus AB/BCUnit 6 – Integration & Accumulation of ChangeTopic: 6.7 6.7 The Fundamental Theorem of Calculus and Definite Integrals