Starter Find the gradient function for each of the following: y = x3 + 7 y = x3 y = x3 – 5 y = x3 + 2 y = x3 – 8 What do you notice? Why do you think this has happened?

Calculus 2 Integration

Calculus 2 As you saw in the starter dy/dx = 3x2 is the gradient function for lots of equations. How many equations have dy/dx = 3x2? Curves of the form y = x3 + c, where c is any number all have dy/dx = 3x2

Integration If we are given a gradient function (dy/dx), integration, is the process of working backwards from this to find the equation of the curve. ∫ means “integrate”

Integration Is the gradient function enough to find the equation of the curve? Suppose dy/dx = 2x what curve has this gradient function? Hint: Think back to the start of differentiation, what differentiates to give you 2x? Are there any others?

Differentiation What is the general rule for differentiating xn? If y = xn then dy/dx = nxn-1 Integration has a similar pattern…..

Integration If dy/dx = xn then ∫ xn dx = xn+1 + c n+1 Explain what the ‘dx’ is for and what the ‘c’ means This just means that you are integrating ‘with respect to x’ as opposed to any other letter This stands for any constant number

How does this work??? The notation can look confusing but when you put it into practice it gets easier You can remember it as: “add one to the power, divide by the new power”

Examples Integrate the following: x7 x2 5 3x3 Hint: Remember “add one to the power and divide by the new power”

What if the equation is more complicated? Suppose dy/dx = 3x2 + 2x + 5 How do we integrate that? You just take each part of the equation at a time to get: ∫ 3x2 + 2x + 5 dx = 3(x3/3) + 2(x2/2) + 5x + c = x3 + x2 + 5x + c Solutions that involve ‘c’ are called indefinite integrals

INTEGRALS Finding the value of c

Finding c Find the general solution to: dy/dx = 6x2 + 2x – 5 y = 6(x3/3) + 2(x2/2) – 5(x) + c y = 2x3 + x2 – 5x + c

Finding c Find the equation of the curve with this gradient function that passes through the point (1,7) So we know the equation is of the form y = 2x3 + x2 – 5x + c How can we find the value of c?

Finding c 7 = 2(13) + (12) - 5(1) + c 7 = 2 + 1 – 5 + c 7 = -2 + c So the equation of the curve is: y = 2x3 + x2 – 5x + 9

Summary To find the equation of a curve: Integrate the gradient function Substitute the coordinates of a point on the curve to find the value of c Write the full equation of the curve