FM Volumes of revolution II around xaxis KUS

FM Volumes of revolution II: around x-axis • KUS objectives BAT Find Volumes of revolution using Integration; rotating around x. Axis Starter: Find these integrals

Notes 1 You can use Integration to find areas and volumes y y dx dx y y a b x In the trapezium rule we thought of the area under a curve being split into trapezia. To simplify this explanation, we will use rectangles now instead The height of each rectangle is y at its x-coordinate The width of each is dx, the change in x values So the area beneath the curve is the sum of ydx (base x height) The EXACT value is calculated by integrating y with respect to x (y dx) a b x For the volume of revolution, each rectangle in the area would become a ‘disc’, a cylinder The radius of each cylinder would be equal to y The height of each cylinder is dx, the change in x So the volume of each cylinder would be given by πy 2 dx The EXACT value is calculated by integrating y 2 with respect to x, then multiplying by π. (πy 2 dx)

Notes 2 y y This would be the solid formed a b x Imagine we rotated the area shaded around the x -axis What would be the shape of the solid formed? x

NOW DO Ex 4 A

• KUS objectives BAT Find Volumes of revolution using Integration; rotating around x. Axis self-assess One thing learned is – One thing to improve is –

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