You can now download NCERT Solutions Class 12 Maths Chapter 8 PDF available at the official website of Vedantu. Our Application of Integrals Class 12 NCERT Solutions is made in an easily understandable manner by the subject matter experts of our team. The teachers are well-versed with the current CBSE syllabus, and their solutions are designed based on that. With these NCERT Solutions for Class 12 Maths Chapter 8, you would be able to quickly revise all the salient points of the chapter and remember all the important formulas.
Ch 8 Maths Class 12 is based on the application of integrals which can be a tricky topic for many of you. In order to master it, you need support from experts who have a thorough knowledge of this topic, and thatâ€™s what our team is well equipped with. On top of the ease of access to quality NCERT Solutions for Class 12 Maths Chapter Application of Integrals on our website, you also get constant support from our subject matter experts in case you get any problem on any topic.
Having easy access to Maths solutions can give the required boost needed by students in todayâ€™s hectic lifestyle. With the readily available NCERT Solutions Class 12 Maths Chapter 8 PDF download at our website, students can revise their syllabus on the go.
In the introduction of Chapter 8 Class 12th Maths, you will be reminded of all the geometrical formulae you learned in the previous chapters. You would recall how these geometric formulae for calculating areas of triangles, rectangles, trapeze, etc. are the basis on which mathematics is applied to real-life problems. However, they are not able to determine areas enclosed by curves. That is where integral calculus comes into the picture.
You will rekindle what you learned about areas bounded by a curve in the previous chapter where definite integrals were calculated as the limit of a sum. In this chapter, you would use those concepts to find out areas between lines and arcs of circles, parabolas, simple curves and ellipses.
AOI Class 12 NCERT Solutions have many solved examples and a variety of questions neatly divided into different exercises. You will get rigorous practice with the set of questions presented in this chapter. Some of the prominent parts discussed here are:
Elementary area i.e. the area located between any arbitrary positions
The area under two curves
The area that is bounded by a curve and a line
Integrals of trigonometric identities
This section of Chapter 8 Class 12 Maths begins with a recap of how definite integrals are calculated as the limit of a sum and the fundamental theorem of calculus. You will then learn how to find the area bounded by the curves. So, if the curve is y = f(x) then to find the area bounded by the curve, the x-axis, and 2 points x=p1 and x=p2 on the x-axis you can consider it as a large number of vertical strips. After this, you would understand how to calculate individual areas of these thin strips. Suppose the height of the strip is y and width is dx then the area of one strip is expressed as dA = ydx, here y = f(x). This area is termed as an elementary strip. The total area can thus be calculated by integrating these elementary strips:
A = \[\int_{p1}^{p2}\] dA = \[\int_{p1}^{p2}\] ydx = \[\int_{p1}^{p2}\] f(x) dx
In case the position of the curve is below the x-axis, the area will come out to be negative, but we take only the absolute value of the integral.
You may learn how to calculate the area of a region which is bounded by a curve and a line. One could consider either vertical or horizontal strips to calculate the area of the region. So if the curve is y = f(x) and the equation of the line is g = ax + k, we can see there are 2 cases; one is the area under the curve, and the other is the area between 2 curves, depending on how many points the line intersects the curve at. For a line intersecting the curve at 2 points, a general formula to find the region between the line and the curve can be given as:
A = \[\int_{p1}^{p2}\] [g(x) - f(x)]dx. Here g(x) is the equation of the line, f(x) is the equation of the curve, and p1, p2 are points of intersection of the curve with the straight line.
In this unit of Application of Integrals Class 12 chapter, you would learn how to calculate area between 2 curves by dividing the common region into elementary areas of vertical strips. So if equation of one curve is y = f(x) and equation of the other curve is y = g(x) and we know that f(x) >= g(x), the elementary area can be given as: dA = [f(x) â€“ g(x)]dx where dx is the width of the strip. The total area is thus calculated by the following integral:
A = \[\int_{p1}^{p2}\] [f(x) - g(x)]dx. Here p1 and p2 are the 2 points of intersections of the curve on the x-axis.
To score well in CBSE and also crack many competitive exams like NEET and IIT, one needs to have a solid base in Mathematics. The NCERT books present you with challenging questions which better your analytical abilities and give you ample exposure to all kinds of questions that can come in any of these exams. Our Application of Integrals Class 12 Solutions is in line with the latest CBSE curriculum and is extremely beneficial for your preparation for the following reasons:
The quality of solutions is impeccable as they are designed by teachers with immense experience in the subject matter.
You get to revise all the key points of a chapter in very less time.
The solutions help you manage your stress and time during exams.
Class 12 Maths Ch 8 NCERT Solutions would help you clear your root level concepts.
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