# Area under a curve

A collection of resources exploring how to find the area under a curve: Integration

### Integrating a Square Root Function

These videos from Casio explain how to integrate a square root function in order to find the area bounded by the curve and the positive x and y axes.

The graphical solution explains how to use the graphic calculator to verify the solution by drawing the graph of the function and using the function of calculator in order to find the required area.

### Finding the Minimum of a Curve and Evaluating a Definite Integral

These videos from Casio explain how to find the coordinates of the minimum point of y = 8x + 1/x, for x > 0 and how to find the area between the curve, the x axis and the lines x = 1 and x = 6 using integration.

The graphical solution explains how to use the graphic calculator to draw the required graph and find the minimum value. The functions on the calculator are then used to find the required area and verify the mathematical solution.

### Using Partial Fractions to Evaluate the Definite Integral of an Algebraic Fraction Example One

The first part of this video from Casio explains how to express an algebraic fraction in partial fractions. The second part of the video uses the result of the first part of the question in order to find the value of an integral between limits expressing the answer as a single logarithm.

Graphical calculator solution shows how a graphical calculator can be used to verify the solution. The steps show how to integrate a function between limits using the functions in the graphical calculator.

### Using Partial Fractions to Evaluate the Definite Integral of an Algebraic Fraction Example Two

These videos from Casio explain how to express a function in partial fractions where the three denominators are linear.

The graphical solution explains how to use the graphic calculator to verify the solution found in the mathematic solution.

### Risps: Rich Starting Points for A Level Mathematics

This NRICH maths activity will give students insights into differentiation, integration and the relationships between the two without needing to get involved with technical manipulations.

It would be well suited to use as an introduction or summary to differentiation or integration. It is very good for giving intuitive meaning to the procedures and features of integration and differentiation, so would suit students with a range of technical skills.

The accurate charts can also be used as a problem involving fitting curves to equations. They can also be used to practice numerical integration

### Differentiating and Integrating Fractional and Negative Powers C4

In this activity students are asked to write expressions of x in index form. There are sets of matching cards in which students match the function with the differential. There is also a set of cards displaying integrals to be matched to funtions.

### Thinking Questions

This thinking question from Susan Wall presents students with an integration problem and the answer. Students are required to add correct limits to the question in order to make the solution correct.

This example gives solutions in terms of natural logs. The idea can be used for other integrals. Students can also be asked to make their own equation for peers to attempt.

### Risps for AS Level Core: Integration

Two RISP activities designed for students to explore or consolidate ideas about integration.

**Introducing e** requires students to use a graphing package to explore a variety of functions of the form y equals x to the power of n and attempt to find the value for k for which the area under the graph between 0 and k is exactly one. Students are then asked to explore different properties of this family of functions leading to a definition of the exponential function.

**The answer’s 1**: what’s the question? gives students a number of graphs containing shaded areas enclosed by two functions. Given that the enclosed area has a value of one, students are asked to find the equations of the functions.

### Integration

These resources which cover aspects of integration and are suitable for students studying mathematics at A Level, as well as those students for whom mathematics is an integral part of their course. Some of the topics covered include **integration as the reverse of differentiation**, **integration by parts**,** integration by substitution **and **finding areas by integration.**

Comprehensive notes, with clear descriptions, for each resource are provided, together with relevant diagrams and examples. Students wishing to review, and consolidate, their knowledge and understanding of integration will find them useful, as each topic includes a selection of questions to be completed, for which answers are provided.

### Risps for A2 Level Core: Integration 2

This RISP activity, **The answer's 1**: what's the question? gives students graphs containing shaded areas enclosed by two functions. Examples of a straight line and a quadratic graph, a cubic graph and an exponential graph are used. Given that the enclosed area has a value of one, students are asked to find the functions. The RISP can also be used to consolidate work on the Trapezium Rule, and to investigate Volumes of Revolution.