# Integration

**AS level **

- Know and use the Fundamental Theorem of Calculus
- Integrate x
^{n}(excluding n = -1), and related sums, differences and constant multiples - Evaluate definite integrals; use a definite integral to find the area under a curve

**A level **

- Integrate e
^{kx}, 1/x, sin(kx), cos(kx) and related sums, differences and constant multiples - Evaluate definite integrals; use a definite integral to find the area under a curve and the area between two curves
- Understand and use integration as the limit of a sum
- Carry out simple cases of integration by substitution and integration by parts; understand these methods as the inverse processes of the chain and product rules respectively (Integration by substitution includes finding a suitable substitution and is limited to cases where one substitution will lead to a function which can be integrated; integration by parts includes more than one application of the method but excludes reduction formulae)
- Integrate using partial fractions that are linear in the denominator
- Evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions (Separation of variables may require factorisation involving a common factor)
- Interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution; includes links to kinematics

### Integration

These resources 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.

### Thinking Questions

This resource from Susan Wall contains a number of open–ended questions which explore understanding and allow a variety of approaches. The activity appropriate for this collection is Integration: one problem exploring integration between limits.

Each question is easily accessible but can be extended to make a more complex problem. Students are required to justify their answer and, where possible, generalise their answer. Students require problem solving skills and reasoning skills to tackle the problems; trial and error alone will not be sufficient.

### Integrating a Square Root Function

This resource contains two video tutorials from Casio.

The first video explains 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 integral is found by differentiating an appropriate function and rearranging and evaluating.

The second 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.

### Risps for A2 Level Core: Integration 2

In this RISP activity The answer's 1: what's the question? students are given 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.

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

This resource contains two video tutorials from Casio.

The first explains 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 second explains how to use the graphic calculator to draw the required graph and find the minimum value. To find the required area, the scales are reset to ensure the required area can be viewed. The functions on the calculator are then used to find the required area and verify the mathematical solution.

### Integration

This series of excel sheets look at the area of regions bounded by linear and quadratic graphs.

The first interactive sheet calculates the area under the line y = mx + c between limits A and B. The effect on the area of adjusting the value of the gradient, y-intercept and upper and lower limits can then be seen.

The next interactive sheet is similar to the first, this time calculating the area of the region bounded the line y = mx + c and the y-axis between limits P and Q.

The next two interactive sheets repeat the processes of calculating area, this time for quadratic functions. Again the variables used, including the coefficients of the equation, can be adjusted.

The final two interactive sheets calculate the area of a region bounded by two functions. The first finds the shaded area between two quadratic functions and the second finds the area of the region bounded by a parabola and a straight line.

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

In these Casio videos, how to express an algebraic fraction in partial fractions in explained, as is how a graphical calculator can be used to verify the solution.

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

These two Casio videos explain how to express a function in partial fractions where the three denominators are linear and how to use the graphic calculator to verify the solution.