How do you find the volume of the solid generated by revolving the given region about the y axis?

Answer: The volume of a solid rotated about the y-axis can be calculated by V = π∫ d c [f(y)] 2 dy .

How do you find volume using disk method?

To find the volume of a disk, first identify the radius function, which is the difference between the original curve and the axis of revolution. Square the radius function and multiply it by pi, and then take the integral of this result from one endpoint to another .

What is the formula for the volume of a solid shell method?

The shell method calculates the volume of the full solid of revolution by summing the volumes of these thin cylindrical shells as the thickness Δ x \Delta x Δx goes to 0 0 0 in the limit: V = ∫ d V = ∫ a b 2 π x y d x = ∫ a b 2 π x f ( x ) d x .

What is the formula for the volume of a revolving solid?

The 3-D model of the solid of revolution. FORMULAS: V= ∫Adx , or respectively ∫Ady where A stands for the area of the typical disc. and r=f(x) or r=f(y) depending on the axis of revolution.

What is the disk method of rotation?

The disk method is used when the axis of revolution is the boundary of the plane region and the cross-sectional area is perpendicular to the axis of revolution . This method is used to find the volume by revolving the curve y=f(x) y = f ( x ) about x -axis and y -axis.

When to use disc method?

If you have a function y=f(x) and you rotate it about the x axis , you should use disk (or ring, same thing in my mind). If you rotate y=f(x) about the y axis, you should use shell. Of course, you can always use both methods if you can find the inverse of the function.

How to calculate shell volume?

How do you calculate the volume of a shell? The volume of a cylindrical shell is 2pi*rh where r is the radius of the cylinder, and h is the height of the cylinder.

How to find the volume of a solid?

Volume of Solids Formula Volume = l×b×h. Total surface area = 2(lb+bh+lh) Length of diagonal of cuboid = √(l 2 +b 2 +h 2 )

How do you calculate volume method?

Height × width × length= volume If the height, width and length are measured in cm, the answer will be cm³.

How do you determine the volume of the solid when the figure below is revolved around the Y axis?

To find the volume of a solid of revolution about the y-axis, use the axis of revolution. The representative rectangle is shown in the figure. The outer and inner radil of this sold are f(y)= and r(y)=yf respectively. According to the washer method, Volume =v=x∫cd([R(y)]2−[r(y)]2)dy .

How do you find the surface area of the solid generated by revolving?

Surface Area of a Surface of Revolution Surface Area=∫ba(2πf(x)√1+(f′(x))2)dx . Surface Area = ∫ a b ( 2 π f ( x ) 1 + ( f ′ ( x ) ) 2 ) d x . Surface Area=∫dc(2πg(y)√1+(g′(y))2)dy. Surface Area = ∫ c d ( 2 π g ( y ) 1 + ( g ′ ( y ) ) 2 ) d y .

Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about each given line. x^2 / 3+y^2 / 3=a^2 / 3, a>0 (hypocycloid) (a) the x-axis (b) the y-axis

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Solutions

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Solutions ${viewAllHTML}

`); let questionUrl = "/questions/xxx/"; let askUrl = "/ask/question/xxx/"; solution_search_result.forEach((elem) => { let url = ('course' in elem)?askUrl:questionUrl; let solution_type = ('course' in elem)?'ask':'question'; let subtitle = ('course' in elem)?(elem.course??""):(elem.book ?? "")+" "+(elem.chapter?"Chapter "+elem.chapter:""); solutions_section.find('#whiletyping-solutions').append(` ${elem.text} ${subtitle} `); }); $('#search-solution-options').empty(); if (Array.isArray(solution_search_result) && solution_search_result.length>0){ $('#search-solution-options').append(solutions_section); } MathJax.typesetPromise([document.getElementById('search-solution-options')]); } } function build_textbooks() { $('#search-pretype-options').empty(); $('#search-pretype-options').append($('#search-solution-options').html()); if (Array.isArray(textbook_search_result)) { var books_section = $(`

Textbooks View All

`); let searchUrl = "/books/xxx/"; textbook_search_result.forEach((elem) => { books_section.find('#whiletyping-books').append(` ${elem.title} ${ordinal(elem.edition)} ${elem.author} `); }); } if (Array.isArray(textbook_search_result) && textbook_search_result.length>0){ $('#search-pretype-options').append(books_section); } } function build_popup(first_time = false) { if ($('#search-text').val()=='') { build_pretype(); } else { solution_and_textbook_search(); } } var search_text_out = true; var search_popup_out = true; const is_login = false; const user_hash = null; function pretype_setup() { $('#search-text').focusin(function() { $('#search-popup').addClass('show'); resize_popup(); search_text_out = false; }); $( window ).resize(function() { resize_popup(); }); $('#search-text').focusout(() => { search_text_out = true; if (search_text_out && search_popup_out) { $('#search-popup').removeClass('show'); } }); $('#search-popup').mouseenter(() => { search_popup_out = false; }); $('#search-popup').mouseleave(() => { search_popup_out = true; if (search_text_out && search_popup_out) { $('#search-popup').removeClass('show'); } }); $('#search-text').on("keyup", delay(() => { build_popup(); }, 200)); build_popup(true); let prevbookUrl = `/search/pretype_books/`; let prebooks = null; try { prebooks = JSON.parse(localStorage.getItem('PRETYPE_BOOKS_'+(is_login?user_hash:'ANON'))); }catch(e) {} if (prebooks && 'previous_books' in prebooks && 'recommended_books' in prebooks) { if (is_login) { previous_books = prebooks.previous_books; recommended_books = prebooks.recommended_books; if (prebooks.time && new Date().getTime()-prebooks.time<1000*60*60*6) { build_popup(); return; } } else { anon_pretype(); return; } } $.ajax({ url: prevbookUrl, method: 'POST', data:{csrfmiddlewaretoken: "mQElkbZZbbY0nWwoIEUjrcTLNOzrdQyMqrjG1d5pWX8H4PhqR63CzwrIeqr6aFA6"}, success: function(response){ previous_books = response.previous_books; recommended_books = response.recommended_books; if (is_login) { localStorage.setItem('PRETYPE_BOOKS_'+user_hash, JSON.stringify({ previous_books: previous_books, recommended_books: recommended_books, time: new Date().getTime() })); } build_popup(); }, error: function(response){ console.log(response); } }); } $( document ).ready(pretype_setup); $( document ).ready(function(){ $('#search-popup').on('click', '.search-view-item', function(e) { e.preventDefault(); let autoCompleteSearchViewUrl = `/search/autocomplete_search_view/`; let objectUrl = $(this).attr('href'); let selectedId = $(this).data('objid'); let searchResults = []; $("#whiletyping-solutions").find("a").each(function() { let is_selected = selectedId === $(this).data('objid'); searchResults.push({ objectId: $(this).data('objid'), contentType: $(this).data('contenttype'), category: $(this).data('category'), selected: is_selected }); }); $("#whiletyping-books").find("a").each(function() { let is_selected = selectedId === $(this).data('objid'); searchResults.push({ objectId: $(this).data('objid'), contentType: $(this).data('contenttype'), category: $(this).data('category'), selected: is_selected }); }); $.ajax({ url: autoCompleteSearchViewUrl, method: 'POST', data:{ csrfmiddlewaretoken: "mQElkbZZbbY0nWwoIEUjrcTLNOzrdQyMqrjG1d5pWX8H4PhqR63CzwrIeqr6aFA6", query: $('#search-text').val(), searchObjects: JSON.stringify(searchResults) }, dataType: 'json', complete: function(data){ window.location.href = objectUrl; } }); }); });

Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about each given line. x^2 / 3+y^2 / 3=a^2 / 3, a>0 (hypocycloid) (a) the x-axis (b) the y-axis | Numerade (2024)

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How do you find the volume of the solid generated by revolving the given region about the y axis? ›

How do you determine the volume of the solid when the figure below is revolved around the Y axis? ›