Area under a curve

Figure 5 shows a sketch of the curve C with equation

The region R, shown shaded in Figure 5, ounded by C, the line with equation x = 1,

Use the substitution x = 2 sinu to show that the area of R is given by

where a, b and k are constants to be found.

Hence, using algebraic integration, find the exact area of R. Give your answer in simplest form.

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 5 shows a sketch of part of the curve with equation y = f (x), where

The curve

• intersects the x ‑axis at –1 and 1

• has minimum turning points at P and Q

as shown in Figure 5.

Use calculus to find the exact coordinates of P.

Using the substitution x = tanθ show that

where α and β are constants to be found. The finite region R, shown shaded in Figure 5, is bounded by the x‑axis and the curve.

Use algebraic integration to find the area of R.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 3 shows a sketch of part of the curve with equation

The curve crosses the x‑axis at the point A.

Verify that the x coordinate of A is 4

The line l₁ is the tangent to the curve at A.

Use calculus to show that an equation of line l₁ is

12 x + y = 48

The line l₂ has equation y = 8x The region R, shown shaded in Figure 3, is bounded by the curve, the line l₁ and the line l₂

Use algebraic integration to find the exact area of R.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 5 shows a sketch of part of the curve C with equation

The finite region R, shown shaded in Figure 5, is bounded by

• the curve C

• the line with equation x = 1

• the x-axis

Find the exact area of R, giving your answer in the form

where A and B are rational numbers to be found.

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

Figure 3 shows a sketch of part of a curve with equation

The region R, shown shaded in Figure 3, is bounded by the curve and the x-axis.

Figure 2 shows a sketch of part of the curve with equation

The finite region R, shown shaded in Figure 2, is bounded by the curve, the line with equation x = 2, the x-axis and the line with equation x = 4

The table below shows corresponding values of x and y, with the values of y given to 4 decimal places.

Use the trapezium rule, with all the values of y in the table, to obtain an estimate for the area of R, giving your answer to 3 significant figures.

Use algebraic integration to find the exact area of R, giving your answer in the form

where a, b and c are integers to be found.

In this question you should show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Figure 2 shows a sketch of part of the curve C with equation

The point P(5, –13) lies on C The line l is the tangent to C at P

Use differentiation to find the equation of l, giving your answer in the form y = mx + c where m and c are integers to be found.

Hence verify that l meets C again on the y-axis. The finite region R, shown shaded in Figure 2, is bounded by the curve C and the line l.

Use algebraic integration to find the exact area of R.

In this question you should show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 2 shows a sketch of part of the curve C with equation

The point P(5, –13) lies on C

The line l is the tangent to C at P

Use differentiation to find the equation of l, giving your answer in the form y = mx + c where m and c are integers to be found.

Hence verify that l meets C again on the y-axis.

The finite region R, shown shaded in Figure 2, is bounded by the curve C and the line l.

Use algebraic integration to find the exact area of R.

Figure 2 shows a sketch of part of the curve with equation y = x (x + 2)(x – 4).

The region R₁ shown shaded in Figure 2 is bounded by the curve and the negative x-axis.

Show that the exact area of R₁ is

The region R₂ also shown shaded in Figure 2 is bounded by the curve, the positive x-axis and the line with equation x = b, where b is a positive constant and 0 b 4

Given that the area of R₁ is equal to the area of R₂

verify that b satisfies the equation

The roots of the equation 3b² – 20b + 20 = 0 are 1.225 and 5.442 to 3 decimal places. The value of b is therefore 1.225 to 3 decimal places.

Explain, with the aid of a diagram, the significance of the root 5.442

Explain why you can deduce that q = 4

Figure 4 shows a sketch of part of the curve C. The region R, shown shaded in Figure 4, is bounded by the curve C, the x-axis and the line with equation x = 3

Show that the exact value of the area of R is a ln 2 + b ln 3, where a and b are rational constants to be found.

Figure 2 shows a sketch of part of the curve C with equation y xlnx, x 0

The line l is the normal to C at the point P(e, e)

The region R, shown shaded in Figure 2, is bounded by the curve C, the line l and the x-axis.

Show that the exact area of R is Ae² B where A and B are rational numbers to be found.

Figure 3 shows a sketch of part of the curve with equation

The finite region S, bounded by the x-axis and the curve, is shown shaded in Figure 3.

Find

Hence find the area of S.

Complete the table below, giving the missing value of y to 3 decimal places.

Figure 1 shows the region R which is bounded by the curve with equation y = , the x-axis and the lines x = 0 and x = 3

Use the trapezium rule, with all the values of y from your table, to find an approximate value for the area of R.

Use your answer to part (b) to find an approximate value for

giving your answer to 2 decimal places.

Figure 3 shows a sketch of part of the curve C with equation y = x (x + 4)(x – 2) The curve C crosses the x-axis at the origin O and at the points A and B.

Write down the x-coordinates of the points A and B.

The finite region, shown shaded in Figure 3, is bounded by the curve C and the x- axis.

Use integration to find the total area of the finite region shown shaded in Figure 3.

The line with equation y = 10 cuts the curve with equation y = x² + 2x + 2 at the points A and B as shown in Figure 1. The figure is not drawn to scale.

Find by calculation the x-coordinate of A and the x-coordinate of B.

The shaded region R is bounded by the line with equation y = 10 and the curve as shown in Figure 1.

Use calculus to find the exact area of R.

Figure 2 shows the line with equation y = 10 – x and the curve with equation y = 10x – x² – 8 The line and the curve intersect at the points A and B, and O is the origin.

Calculate the coordinates of A and the coordinates of B.

The shaded area R is bounded by the line and the curve, as shown in Figure 2.

Calculate the exact area of R.

The straight line with equation y = x+ 4 cuts the curve with equation y x² + 2x + 24 at the points A and B, as shown in Figure 3.

Use algebra to find the coordinates of the points A and B.

The finite region R is bounded by the straight line and the curve and is shown shaded in Figure 3.

Use calculus to find the exact area of R.