Problem 653
Compute the midspan value of EIδ for the beam shown in Fig. P-653. (Hint: Draw the M diagram by parts, starting from midspan toward the ends. Also take advantage of symmetry to note that the tangent drawn to the elastic curve at midspan is horizontal.)
 

The deflection δ at some point B of a simply supported beam can be obtained by the following steps:
 

Forces acting at some angle from the the coordinate axes can be resolved into mutually perpendicular forces called components. The component of a force parallel to the x-axis is called the x-component, parallel to y-axis the y-component, and so on.
 

Coplanar Parallel Force System
Parallel forces can be in the same or in opposite directions. The sign of the direction can be chosen arbitrarily, meaning, taking one direction as positive makes the opposite direction negative. The complete definition of the resultant is according to its magnitude, direction, and line of action.
 

Resultant of a force system is a force or a couple that will have the same effect to the body, both in translation and rotation, if all the forces are removed and replaced by the resultant.
 

Example 6
What is the area within the curve r² = 16 cos θ?
 

Find the area enclosed by four-leaved rose r = a cos 2θ.
 

Example 3
Find the area inside the cardioid r = a(1 + cos θ) but outside the circle r = a.
 

Example 6
Find each of the two areas bounded by the curves y = x³ - 4x and y = x² + 2x.