(a) Graph the function f(x)=1 /(1+x^2) -2 ⩽x ⩽2 (b) Estimate the area under the graph of f usi

(a) Graph the function f(x)=1 /(1+x^2) -2 ⩽x ⩽2 (b)...

Key Concepts

Refinement of Approximations

Refining approximations involves increasing the number of subintervals (or rectangles) used in a Riemann sum, which typically leads to a more accurate estimate of the area under a curve. As the number of rectangles increases, the approximation converges to the exact value of the definite integral.

Sampling Methods

Sampling methods in the context of Riemann sums refer to the choice of the sample points used to determine the height of each rectangle. Common choices include the right endpoint and the midpoint of each subinterval. These choices affect the approximation's accuracy and are important in numerical integration methods.

Riemann Sums

Riemann sums are a method to approximate the definite integral of a function by dividing the area under a curve into a finite number of rectangles. The sum of the areas of these rectangles approximates the total area under the curve, and the accuracy of the approximation increases with a greater number of partitions.

Area Under a Curve

The area under a curve is a fundamental concept in integral calculus, representing the accumulation of quantities. It is calculated as the definite integral of a function over an interval, which geometrically corresponds to the region bounded by the function, the x-axis, and the vertical lines at the interval's endpoints.

Graphing Functions

Graphing functions involves plotting the coordinates of a function on a coordinate plane, analyzing its shape, behavior, and key features like symmetry, intercepts, and asymptotes. This helps in visualizing the function, which is essential for understanding its behavior over a given interval.

Transcript

00:01 Okay so in this question we are asked to plot this graph so first about in order to plot this graph we need to find that hack curve of that function so we put or plot the x in this rectangle this negative 2 -2 and find the value of fx so when we start at negative 2 it will be 0 .2 when it is just 1 it will be 0 .5 0 it will be 1 when it is 1 is 0 .5 again to 0 .2 again so this negative negative 1 and 2 and this is 1 so when x equals to 0 the hour the curve is at 1 and when it is negative 1 is at the 0 .5 5.

01:23 And 1 it is 0 .5 when it is 2 it is 0 .2 and negative 2 is 0 .2 again so this is the curve or this is the graph of our function given function and on the part b we are asked to estimate the area under the this curve making the four approximate rectangles so we have the curve over here this is negative to approach 2 and this is negative 2 and this is negative 1 and this is 1 so so in the first part of this part b we are asked to use the right in points so yeah if you use the right in points then our rectangle four rectangle will be the first one will be like this the second one will start from the zero and in a negative 1 and the third one will start from one and emit the curve and we draw the horizontal line from that point to the y -axis and similarly from the two so these are the four rectangles so these are the four rectangles and which will help us to find the area underneath the curve so the first thing is we know that in the area of the rectangle is a length times the width of that triangle rectangles so it's our and the base is the constant that is one and we just need to find the length of that rectangle and whenever this point the core rectangle meets the call that is our length so our total area will be area of rectangle one plus area of rectangle two plus so a one will be the width is one and the length is so this length is 0 .0 .5 because earlier we find the point on this curve by putting the value of x on the fx and that's the way we find the point y on the that curve and that is our length of our rectangles.

05:12 So the area of rectangle 2 is we have the length 1 and width 1.

05:21 And for the area of rectangle of number 3, we have the length 1 and the width 1 and and for the last triangle on the right -hand side, we have length equals to 0 .2 and width equals to 1 the basis same for all the rectangles and 0 .5 plus 1 plus and 0 .5 plus 0 .2 when we add up this all 2 it will wind up 2 .2 so this is our area of a area underneath the curve using the writing points to make the four rectangles to estimate the area underneath the curve.

06:25 And again the part b says that use the midpoints to draw the triangle and estimate it a rectangle.

06:51 So i will first draw my curve.

07:06 So negative two.

07:06 So we have to find the midpoint of all of this and that will help us to make the rectangles to estimate the area under this curve.

07:20 So the midpoint in between negative 2 and negative 1 is negative 1 .5.

07:25 So this is our midpoint and this is how our first rectangle looks like.

07:35 And the next midpoint is 0 .5.

07:41 And this is our second rectangle and again it is 0 .5 from 0 to 1 this is our second rectangle and it is 1 .5 again so my rectangles are a little bit messy because i have to draw it in the whiteboard and it is not that much clear but if you do it in our paper it will a lot clear than this so what i'm trying to do over here is just give you the concept of how you would make those triangles and quite estimate the area underneath that curve so similarly we have to find the area of these rectangles and that will estimate the area underneath the curve so you know one a 2, a3, again we are the basis 1 and it is still remain constant and the length of this triangle is f negative 1 .5 the value of a function when it is at the midpoint.

09:44 So if you put the negative 1 .5 in the function then we will get the value 0 .3 1...

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