Application: Area under and between Curve(s) Volume Generated Integration: Part 3 Application: Area under and between Curve(s) Volume Generated

Integration: Application Area Volume Work Pressure

Definite Integration: Refresh Use your calculator to double check your answers

Application : Area Under a Curve Shaded area that is bordered by y=f(x), x=a, x=b and x-axis is Area above x-axis is +ve Area below x-axis is –ve and would need to |-ve| y= x=a x=b A

Application: Area Under a Curve

Application: Area Under a Curve

Application: Area Under a Curve - Steps Step-by-Step Identify the function. Sketch the graph to visualise (if needed) Visualise and shade the area in question Identify the border(s) for the area Perform definite Integration, accordingly. If –ve prediction, absolute the value using the | | sign Add together the area (s) (if needed) Note in unit2 (An are MUST be +ve)

Area Under a Curve: Example

Application: Area Between Curves If f and g are continuous with f(x) => g(x) throughout [a , b], then the area of the region between the curves y = f(x) and y = g(x) from a to b is the integral of [f – g] from a to b.

Area Between Curves

Area Between Curves - Steps Step 1: Sketch the curves and note the intersecting points Step 2: Find the limits of integration by finding the intersecting points (y = y). Step 3: Write a formula for f(x) – g(x) (depending on the which is the top curve and bottom curve). Simplify it. Step 4: Integrate f(x) – g(x) of Step3 from a to b. The value obtained is the area (units2).

Intersecting curves example y 2 -1 1 -2 3 y = 2 - x2 y = - x x

Areas between curves The region runs from x = -1 to x = 2. The limits of integration are a = -1, b =2. The area between the curves is

Area Between Curves Find area enclosed between y= - x2 + 5x and y=2x Find area between y = x 2 - 2x + 2 and y=-x 2 + 6 Find area between y = x 2 – 2x+ 3 and y = 2x3 -12x

Non-intersecting curves example /4 1 y = sin x y = sec2x x-axis 2 (x, g(x)) y-axis (x, f(x)) x

Area between a curve and a line (trigonometric function) y =1  y =sin2 x /2 x-axis y-axis 1

Area Between Curves Find area between y=x2 and y=-x2 [3 , 6] Find area enclosed between y= - x2 + 5x and y=8x+10 [-1 ,0] Find area between y = sin(x) and y=1 [3 , 6] Find area between y = cos(x) and y=-1[-8 , -7]

Integration with respect to y-axis If a region’s bounding curves are described by functions of y, are would be easier calculated horizontal instead of vertical and the basic formula has y in place of x Area under the curve, you would need to modify the equation to be in terms of y Formula for Area between curves

Integration with respect to y-axis Find the area that is bounded by x = y + 2 , x = y2 and by the x-axis.

Integration with respect to y-axis The region’s right-hand boundary is the line x = y + 2, so f (y) = y + 2 The left-hand boundary is the curve x = y2, so g (y) = y2. The lower limit of integration is y = 0. We find the upper limit by solving x = y + 2 and x = y2 simultaneously for y: y = -1, y = 2

Y-axis: Examples Find area under the curve for x = 8y – y2 from y = 0 to y = 7 Find area between x=y2 and y=2x – 1 Find area enclosed between x = (y - 2)2 and x=1 Find area between x = 4y – 2y2, y=2x - 1

Application: Volume Generated (Disc Method)

Volume Generated - Steps STEP 1: Square the equation i.e. (3x)2 STEP2: Perform steps like Area under/between curves STEP 2.5: Definite Integral as such STEP 3: State in units3

Application: Volume Generated (2 Curves)

Volume Generated: Examples Find volume generated from the curve y = x3 + x2 – 6x [-3 , 0] rotated along the x-axis. Find volume generated between y = 2x2 - 4x + 6 and y = x2 + 2x + 1 for in respect to the x-axis Find volume generated between y=x2 - 2x +1 and y=2x – 1 being rotated along the x-axis Find volume generated between y = - x2 + 5x and y = 2x rotated along the x-axis