Integration (OCR A Level Maths: Pure)

Use calculus to find

Evaluate

Find the equation of the curve passing through the point (4, 64) and given by

Show that

Hence, or otherwise, work out

Given

find the value of the positive constant k.

The diagram below shows part of the graph of

Write down the values of x where .

Show that

Evaluate

Write down the area of the region labelled R.

Given  and that when ,,  show that

Find an equation for y in terms of x, given that when , .

The diagram below shows part of the graph of  .

Find the area of the shaded region labelled R.

Find the x-coordinates of the intercepts of the line with equation and the curve with equation 

Evaluate

The diagram below shows the graphs of and  .
Find the exact area of the shaded region R.

The diagram below shows the graphs of the line and  the curve 

Work out the x-coordinates of the points labelled P, Q and R.

Work out the area of the shaded region.

Use calculus to find

Write down

Use calculus to find the value of

Find the equation of the curve passing through the point (-2, 3) and given by

Using the binomial expansion, or otherwise, show that

Hence, or otherwise, work out

Given

find the value of the constant p, where .

The diagram below shows part of the graph of

Find the area of the shaded region labelled R.

A function,, has second derivative given by

Given that , and find

The diagram below shows part of the graph of .

Find the total area of the two shaded regions.

The line with equation    cuts the curve with equation  at the points P and Q, as shown.

Find the x- and y-coordinates of the points P and Q.

Find the exact area of the region labelled R, giving your answer in the form  , where a and b are integers to be found.

The diagram below shows the graphs of and 

Find the exact area of the shaded region.

Use calculus to find

Use calculus to find the value of

giving your answer correct to 3 significant figures.

Find the equation of the curve passing through the point (4, -8) and given by

Show that

Hence, or otherwise, work out

Given

find the value of the constant q.

The diagram below shows part of the graph of

Find the area of the shaded region labelled R.

Without doing any additional calculation, explain why    must be equal to zero.

A function, , has second derivative given by

Given that and are factors of , find .

The diagram below shows part of the graph of

Find the total area of the two shaded regions.

The diagram below shows the graphs of  and  .

Find the exact area of the shaded region.

The diagram below shows the graphs of  and  .

Find the total area of the shaded regions.