The Fundamental Theorem of Calculus by Slidesgo.pptx

Integral calculus is a powerful tool that can be used to solve a wide range of problems in various fields. Here are some examples:
Area Under a Curve: Integral calculus can be used to find the area under a curve and between the curve and the x-axis. This is particularly useful in physics and engineering for finding the total accumulation of a quantity over time or space.

Volume of Solids: By using integration, one can calculate the volume of solids that have curved surfaces, such as spheres, cones, cylinders, and solids of revolution (generated by revolving a curve around an axis).

Work Done by a Force: In physics, the work done by a force can be calculated by integrating the force with respect to the distance over which the force is applied. This is useful in mechanics for analyzing the work done in compressing or stretching springs, lifting objects, or moving fluids.

Center of Mass: Integral calculus can be used to find the center of mass of a continuous body with a varying density. This is important in engineering and physics for analyzing the stability and motion of objects.

Moment of Inertia: The moment of inertia of an object, which is a measure of its resistance to rotational acceleration, can be calculated using integration. This is crucial in rotational dynamics and engineering for designing structures and machines.

Arc Length: The length of a curve can be found by integrating the differential arc length over the interval of interest. This is useful in various applications, such as road and railway design.

Hydrostatic Pressure and Force: In fluid mechanics, integral calculus can be used to calculate the pressure and force exerted by a fluid on a submerged object or a dam wall.

Probability: In statistics, integration is used to calculate the probability of a continuous random variable falling within a certain range.

Thermodynamics: Integral calculus is used to calculate the change in internal energy, work done, and heat transfer in thermodynamic processes.

Electromagnetism: In physics, integration is used to calculate the electric field or magnetic field generated by continuous charge or current distribution
Differential Calculus:

This branch deals with the concept of the derivative, which represents the rate of change of a function with respect to a variable.
Derivatives are used to find slopes of curves, to optimize functions (find maximum and minimum values), and to solve various real-world problems involving rates of change, such as velocity and acceleration in physics.
The process of finding a derivative is called differentiation.
Integral Calculus:

This branch focuses on the concept of the integral, which represents the accumulation of quantities and can be used to find areas, volumes, and other quantities that can be represented as the limit of a sum.
Integrals are used to solve problems involving the accumulation of change over an interval and can be applied to various fields such as physics, engineering field .

1. The Fundamental Theorem of Calculus

2. Table of contents 01 Antiderivatives/ Indefinite integrals recap 02 Fundamental theorem of calculus

3. 01 Antiderivatives/ Indefinite integrals recap

4. A super important recap Indefinite integration is used to find the antiderivative of functions . You have a function, assume that it is the derivative , and then you try to trace back to what the original function must have been For example, let’s go back to finding the antiderivative of x 2 . The notation we use for this integral is: ∫x 2 dx We would add 1 to the exponent and get 3 for our new exponent . We then have to add a coefficient to the front , which is 1/exponent. In our case, that would be 1/3 So our final answer for the indefinite integral is (1/3)x 3 + c (we always add the plus c when we integrate, unless we are given a point of the function to plug in)

5. 02 Fundamental theorem of calculus

6. A super important theorem Now that we recapped anti-derivatives, let’s revisit the original definition of integrating, which is finding the area under a curve . What if we wanted to find the area under x 2 between the values x = 2 and x = 5? Our integral would now look like this: ∫ x 2 dx This is known as a definite integral since the bounds are specified 5 2

7. A super important theorem Finding the area ∫ f(x)dx=F(b)-F(a) b a 01 You would first need to find the antiderivative of your function 02 Plug in your a and b values 03 You would then subtract F(a) from F(b) to get your area

8. A super important theorem Now let’s apply it to the problem we did in the previous section 5 2 ∫ x 2 dx The antiderivative would be (1/3)(x 3 ) + c. If we plug 5 for x, we get 125/3 + c. If we plug in 2 for x, we get 8/3 + c. When we perform the subtraction, we get 125/3 + c –(8/3 + c) = 125/3 + c – 8/3 – c = 117/3 117/3 is our answer and we solved it with the fundamental theorem of calculus

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