Area in Polar

Today in class, we learned how to find the area of a function in polar. In the rectangular coordinate system, we can find the area of a function by breaking the area into many small rectangles that can then be added together to find the total area. Instead, in the polar format we can find the area of a function by breaking up the area into many sectors that all come out of the origin. The upper and lower bound of the integral are the radial lines between theta = a and theta = b that bind the area you are trying to find.

Think about a radar on a weather map as the line travels in a counter-clockwise motion. This motion is similar to how we find the area.

When solving, think about certain identities that may be useful to help solve the integral - as we did in today's example.

Slopes of Tangent Lines in Polar

Today in class, we learned about how to find the slope of a tangent line to a curve in Polar. Since slope is defined as change in y over change in x, we need to know r and theta in terms of x and y. So we use x = rcos(theta) and y = rsin(theta). We differentiate each with respect to theta and then divide dy/d(theta) by dx/d(theta), similar to what we did in parametrics, because that will give us dy/dx so that we can plug in theta and find the slope of the line tangent at that point. We can also find the points where there are horizontal tangent lines and the slope of the tangent line at the pole for that graph. It can be quite tedious with a LOT of trig. Hope this helps!

Polar Coordinates

Today in class we learned about using polar coordinates. The points for polar coordinates are (r,θ), where r is the directed distance from the origin (otherwise known as a pole), and where θ is the angle measured in standard position. We also learned how to switch from polar to rectangular equations and how to switch from rectangular to polar equations. There are also some examples we did in class to help.

Parametric Equations in a Plane

Today in class we learned about particle motion in a plane using position, velocity, speed and displacement. We learned that the distance traveled by the particle is equal to the arc length. Using the integral of speed, you can find the arc length, which is synonymous to the distance traveled by the particle.

Arc Length

Today in class, we learned about Arc Length. Since we know the length of a line segment is equal to the square root of (change in x squared plus change in y squared), we can multiply that formula by 1 in the form of change in t over change in t to get the formula for arc length. This formula turns out to be the integral from t1 to t2 of the square root of (dv/dt)^2 + (dy/dt)^2. We can now use this formula to find the arc length of an integral for parametric equations.

Second Derivative with Parametric Equations

Today in class we built upon the knowledge of last weeks lesson (finding the slope of tangent lines to find the slope of lines tangent to parametric equations) by learning an additional step: finding the second derivative of a parametric equation.

Differentiating Parametric equations

Today in class we used our prior knowledge in finding the slope of tangent lines to find the slope of lines tangent to parametric equations. By using the Chain rule, we found that the derivative of y with respect to x equals the derivative of y with respect to t divided by the derivative of x with respect to t (as shown in notes). Using prior knowledge from the fall semester, it is easy to apply the same rules to this concept with a new variable t. Answers may be left in the y-intercept form when solving for the equation of the line tangent to the curve.