Integration | Perplex

Integration

Video Reviews

Watch comprehensive video reviews for Integration, designed for final exam preparation. Each video includes integrated problems you can solve alongside detailed solutions.

Not your average video:

Interactive Problems: Solve problems alongside the video with step-by-step guidance and detailed solutions.

Exam Preparation: Complete unit reviews designed for final exam preparation with all key concepts covered systematically.

Expert Teaching: High-quality instruction from Perplex co-founder James Mullen with clear explanations, worked examples, and exam tips.

Not your average video:

Interactive Problems: Solve problems alongside the video with step-by-step guidance and detailed solutions.

Exam Preparation: Complete unit reviews designed for final exam preparation with all key concepts covered systematically.

Expert Teaching: High-quality instruction from Perplex co-founder James Mullen with clear explanations, worked examples, and exam tips.

Definite integralsKinematicsHL

The video will automatically pause when it reaches a problem.

Area between curve and x-axis

SL 5.11

In general, the area enclosed between a curve and the x-axis is given by

A=∫ab∣f(x)∣dx📖

since any region below the x-axis has f(x)<0, but area must always be positive.

Powered by Desmos

This can be done with technology, or by splitting the integral into parts - where f is positive and where f is negative:

A=∫amf(x)dx+∫mb−f(x)dx🚫

Example

Find the area, shaded in the diagram below, enclosed between the curve of y=x3−x and the x-axis.

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We see from the graph that y is positive for −1<x<0 and negative for 0<x<1:

A =∫−11∣∣x3−x∣∣dx =∫−10(x3−x)dx+∫01−(x3−x)dx =[4x4−2x2]−10−[4x4−2x2]01 =[0−(41−21)]−[(41−21)−0] =21

Area between curve and x-axis

SL 5.11

In general, the area enclosed between a curve and the x-axis is given by

A=∫ab∣f(x)∣dx📖

since any region below the x-axis has f(x)<0, but area must always be positive.

Powered by Desmos

This can be done with technology, or by splitting the integral into parts - where f is positive and where f is negative:

A=∫amf(x)dx+∫mb−f(x)dx🚫

Example

Find the area, shaded in the diagram below, enclosed between the curve of y=x3−x and the x-axis.

Powered by Desmos

We see from the graph that y is positive for −1<x<0 and negative for 0<x<1:

A =∫−11∣∣x3−x∣∣dx =∫−10(x3−x)dx+∫01−(x3−x)dx =[4x4−2x2]−10−[4x4−2x2]01 =[0−(41−21)]−[(41−21)−0] =21